Deterministic Methods for the Boltzmann Equation
Transcript of Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Deterministic Methods for theBoltzmann Equation
A. Narayan and A. Klockner
AM281.1 Convection-Dominated ProblemsOctober 15, 2007
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Outline
1 IntroductionThe EquationCollisions
2 Time-Step SplittingOperator splittingRunge-Kutta
3 TransportFlux-Balance MethodsSemi-Lagrangian Methods
4 Collision OperatorDiscretizationBobylev-Rjasanow’s Integral Transform MethodPareschi-Russo’s Spectral MethodMouhot-Pareschi’s Sub-N6 Method
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 3: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/3.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Outline
1 IntroductionThe EquationCollisions
2 Time-Step SplittingOperator splittingRunge-Kutta
3 TransportFlux-Balance MethodsSemi-Lagrangian Methods
4 Collision OperatorDiscretizationBobylev-Rjasanow’s Integral Transform MethodPareschi-Russo’s Spectral MethodMouhot-Pareschi’s Sub-N6 Method
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 4: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/4.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann Equation:
Is an equation of Statistical Mechanics.
Describes a rarefied gas.
Rarefied gas?→ Enough space that particles in the same volume element mayhave differing velocities.
Contrast: Continuum Mechanics→All particles in a volume element are moving with constantvelocity.(Navier-Stokes Equations)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 5: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/5.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann Equation:
Is an equation of Statistical Mechanics.
Describes a rarefied gas.
Rarefied gas?→ Enough space that particles in the same volume element mayhave differing velocities.
Contrast: Continuum Mechanics→All particles in a volume element are moving with constantvelocity.(Navier-Stokes Equations)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 6: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/6.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann Equation:
Is an equation of Statistical Mechanics.
Describes a rarefied gas.
Rarefied gas?
→ Enough space that particles in the same volume element mayhave differing velocities.
Contrast: Continuum Mechanics→All particles in a volume element are moving with constantvelocity.(Navier-Stokes Equations)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 7: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/7.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann Equation:
Is an equation of Statistical Mechanics.
Describes a rarefied gas.
Rarefied gas?→ Enough space that particles in the same volume element mayhave differing velocities.
Contrast: Continuum Mechanics→All particles in a volume element are moving with constantvelocity.(Navier-Stokes Equations)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 8: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/8.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann Equation:
Is an equation of Statistical Mechanics.
Describes a rarefied gas.
Rarefied gas?→ Enough space that particles in the same volume element mayhave differing velocities.
Contrast: Continuum Mechanics
→All particles in a volume element are moving with constantvelocity.(Navier-Stokes Equations)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 9: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/9.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann Equation:
Is an equation of Statistical Mechanics.
Describes a rarefied gas.
Rarefied gas?→ Enough space that particles in the same volume element mayhave differing velocities.
Contrast: Continuum Mechanics→All particles in a volume element are moving with constantvelocity.
(Navier-Stokes Equations)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 10: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/10.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann Equation:
Is an equation of Statistical Mechanics.
Describes a rarefied gas.
Rarefied gas?→ Enough space that particles in the same volume element mayhave differing velocities.
Contrast: Continuum Mechanics→All particles in a volume element are moving with constantvelocity.(Navier-Stokes Equations)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 11: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/11.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
Density Functions
The Boltzmann equation is solved by a density function f (x , v , t)
f : R3 ×R3 ×R→ R+0 .
Density function: ∫R3×R3
f (v , x , t)dxdv = n(t).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 12: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/12.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
Density Functions
The Boltzmann equation is solved by a density function f (x , v , t)
f : R3 ×R3 ×R→ R+0 .
Density function: ∫R3×R3
f (v , x , t)dxdv = n(t).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 13: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/13.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
Density Functions
The Boltzmann equation is solved by a density function f (x , v , t)
f : R3 ×R3 ×R→ R+0 .
Density function:
∫R3×R3
f (v , x , t)dxdv = n(t).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 14: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/14.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
Density Functions
The Boltzmann equation is solved by a density function f (x , v , t)
f : R3 ×R3 ×R→ R+0 .
Density function: ∫R3×R3
f (v , x , t)dxdv = n(t).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 15: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/15.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann equation is an Integro-PDE.
∂f
∂t+ v · ∇x f =
1
knQ(f , f ), x , v ∈ R3
Also with Forces:∂f
∂t+ v · ∇x f + F · ∇v f =
1
knQ(f , f ), x , v ∈ R3
LHS: particle transport according to v (“Vlasov Equation”)RHS: describes binary collisions between gas particles
Forces may come from: Maxwell’s Equations (Lorentz force),Poisson Equation (electrostatic force), etc. Force PDEs dependon f , making the whole system nonlinear.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 16: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/16.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann equation is an Integro-PDE.
∂f
∂t+ v · ∇x f =
1
knQ(f , f ), x , v ∈ R3
Also with Forces:∂f
∂t+ v · ∇x f + F · ∇v f =
1
knQ(f , f ), x , v ∈ R3
LHS: particle transport according to v (“Vlasov Equation”)RHS: describes binary collisions between gas particles
Forces may come from: Maxwell’s Equations (Lorentz force),Poisson Equation (electrostatic force), etc. Force PDEs dependon f , making the whole system nonlinear.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 17: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/17.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann equation is an Integro-PDE.
∂f
∂t+ v · ∇x f =
1
knQ(f , f ), x , v ∈ R3
Also with Forces:
∂f
∂t+ v · ∇x f + F · ∇v f =
1
knQ(f , f ), x , v ∈ R3
LHS: particle transport according to v (“Vlasov Equation”)RHS: describes binary collisions between gas particles
Forces may come from: Maxwell’s Equations (Lorentz force),Poisson Equation (electrostatic force), etc. Force PDEs dependon f , making the whole system nonlinear.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 18: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/18.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann equation is an Integro-PDE.
∂f
∂t+ v · ∇x f =
1
knQ(f , f ), x , v ∈ R3
Also with Forces:∂f
∂t+ v · ∇x f + F · ∇v f =
1
knQ(f , f ), x , v ∈ R3
LHS: particle transport according to v (“Vlasov Equation”)RHS: describes binary collisions between gas particles
Forces may come from: Maxwell’s Equations (Lorentz force),Poisson Equation (electrostatic force), etc. Force PDEs dependon f , making the whole system nonlinear.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 19: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/19.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann equation is an Integro-PDE.
∂f
∂t+ v · ∇x f =
1
knQ(f , f ), x , v ∈ R3
Also with Forces:∂f
∂t+ v · ∇x f + F · ∇v f =
1
knQ(f , f ), x , v ∈ R3
LHS: particle transport according to v (“Vlasov Equation”)
RHS: describes binary collisions between gas particles
Forces may come from:
Maxwell’s Equations (Lorentz force),Poisson Equation (electrostatic force), etc. Force PDEs dependon f , making the whole system nonlinear.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 20: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/20.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann equation is an Integro-PDE.
∂f
∂t+ v · ∇x f =
1
knQ(f , f ), x , v ∈ R3
Also with Forces:∂f
∂t+ v · ∇x f + F · ∇v f =
1
knQ(f , f ), x , v ∈ R3
LHS: particle transport according to v (“Vlasov Equation”)RHS: describes binary collisions between gas particles
Forces may come from:
Maxwell’s Equations (Lorentz force),Poisson Equation (electrostatic force), etc. Force PDEs dependon f , making the whole system nonlinear.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 21: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/21.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann equation is an Integro-PDE.
∂f
∂t+ v · ∇x f =
1
knQ(f , f ), x , v ∈ R3
Also with Forces:∂f
∂t+ v · ∇x f + F · ∇v f =
1
knQ(f , f ), x , v ∈ R3
LHS: particle transport according to v (“Vlasov Equation”)RHS: describes binary collisions between gas particles
Forces may come from:
Maxwell’s Equations (Lorentz force),Poisson Equation (electrostatic force), etc. Force PDEs dependon f , making the whole system nonlinear.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 22: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/22.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann equation is an Integro-PDE.
∂f
∂t+ v · ∇x f =
1
knQ(f , f ), x , v ∈ R3
Also with Forces:∂f
∂t+ v · ∇x f + F · ∇v f =
1
knQ(f , f ), x , v ∈ R3
LHS: particle transport according to v (“Vlasov Equation”)RHS: describes binary collisions between gas particles
Forces may come from: Maxwell’s Equations (Lorentz force),
Poisson Equation (electrostatic force), etc. Force PDEs dependon f , making the whole system nonlinear.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 23: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/23.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann equation is an Integro-PDE.
∂f
∂t+ v · ∇x f =
1
knQ(f , f ), x , v ∈ R3
Also with Forces:∂f
∂t+ v · ∇x f + F · ∇v f =
1
knQ(f , f ), x , v ∈ R3
LHS: particle transport according to v (“Vlasov Equation”)RHS: describes binary collisions between gas particles
Forces may come from: Maxwell’s Equations (Lorentz force),Poisson Equation (electrostatic force),
etc. Force PDEs dependon f , making the whole system nonlinear.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 24: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/24.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann equation is an Integro-PDE.
∂f
∂t+ v · ∇x f =
1
knQ(f , f ), x , v ∈ R3
Also with Forces:∂f
∂t+ v · ∇x f + F · ∇v f =
1
knQ(f , f ), x , v ∈ R3
LHS: particle transport according to v (“Vlasov Equation”)RHS: describes binary collisions between gas particles
Forces may come from: Maxwell’s Equations (Lorentz force),Poisson Equation (electrostatic force), etc.
Force PDEs dependon f , making the whole system nonlinear.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 25: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/25.jpg)
Introduction Time-Step Splitting Transport Collision Operator
The Equation
The Boltzmann Equation
The Boltzmann equation is an Integro-PDE.
∂f
∂t+ v · ∇x f =
1
knQ(f , f ), x , v ∈ R3
Also with Forces:∂f
∂t+ v · ∇x f + F · ∇v f =
1
knQ(f , f ), x , v ∈ R3
LHS: particle transport according to v (“Vlasov Equation”)RHS: describes binary collisions between gas particles
Forces may come from: Maxwell’s Equations (Lorentz force),Poisson Equation (electrostatic force), etc. Force PDEs dependon f , making the whole system nonlinear.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 26: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/26.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Knudsen Number
∂f
∂t+ v · ∇x f =
1
knQ(f , f )
What is kn?
→ The Knudsen Number.
kn :=λ
L:=
mean free path
representative physical length scale
kn → 0: Continuum limit.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 27: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/27.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Knudsen Number
∂f
∂t+ v · ∇x f =
1
knQ(f , f )
What is kn? → The Knudsen Number.
kn :=λ
L:=
mean free path
representative physical length scale
kn → 0: Continuum limit.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 28: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/28.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Knudsen Number
∂f
∂t+ v · ∇x f =
1
knQ(f , f )
What is kn? → The Knudsen Number.
kn :=λ
L:=
mean free path
representative physical length scale
kn → 0: Continuum limit.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 29: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/29.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Knudsen Number
∂f
∂t+ v · ∇x f =
1
knQ(f , f )
What is kn? → The Knudsen Number.
kn :=λ
L:=
mean free path
representative physical length scale
kn → 0:
Continuum limit.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 30: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/30.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Knudsen Number
∂f
∂t+ v · ∇x f =
1
knQ(f , f )
What is kn? → The Knudsen Number.
kn :=λ
L:=
mean free path
representative physical length scale
kn → 0: Continuum limit.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 31: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/31.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
The collision term splits into gain and loss.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ] =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v∗)dωdv∗.
Arggh. Too much notation. What are we describing, anyway?
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 32: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/32.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
The collision term splits into gain and loss.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ] =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v∗)dωdv∗.
Arggh. Too much notation. What are we describing, anyway?
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 33: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/33.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
The collision term splits into gain and loss.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ] =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v∗)dωdv∗.
Arggh. Too much notation. What are we describing, anyway?
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 34: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/34.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
The collision term splits into gain and loss.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ] =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v∗)dωdv∗.
Arggh. Too much notation. What are we describing, anyway?
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 35: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/35.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
The collision term splits into gain and loss.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ] =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v∗)dωdv∗.
Arggh. Too much notation. What are we describing, anyway?
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 36: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/36.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
The collision term splits into gain and loss.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ] =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v∗)dωdv∗.
Arggh. Too much notation. What are we describing, anyway?
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 37: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/37.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
The collision term splits into gain and loss.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ] =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v∗)dωdv∗.
Arggh. Too much notation.
What are we describing, anyway?
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
The collision term splits into gain and loss.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ] =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v∗)dωdv∗.
Arggh. Too much notation. What are we describing, anyway?A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 39: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/39.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Binary Collisions
θω
v′ v′∗
v∗
v
Particle P Particle P∗
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 40: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/40.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
Ok, let’s look at this again.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ]f =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v)f (v∗)dωdv∗.
Q is a spatially local operator!
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 41: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/41.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
Ok, let’s look at this again.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ]f =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v)f (v∗)dωdv∗.
Q is a spatially local operator!
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 42: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/42.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
Ok, let’s look at this again.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ]f =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v)f (v∗)dωdv∗.
Q is a spatially local operator!
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 43: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/43.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
Ok, let’s look at this again.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ]f =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v)f (v∗)dωdv∗.
Q is a spatially local operator!
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 44: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/44.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
Ok, let’s look at this again.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ]f =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v)f (v∗)dωdv∗.
Q is a spatially local operator!
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 45: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/45.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
Ok, let’s look at this again.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ]f =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v)f (v∗)dωdv∗.
Q is a spatially local operator!
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 46: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/46.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Term
Ok, let’s look at this again.
Q(f , f ) = Q+(f , f )− L[f ]f
Gain Term:
Q+(f , f ) =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v ′)f (v ′∗)dωdv∗,
Loss Term:
L[f ]f =
∫R3
∫S2
B(|v − v∗|, cos θ)f (v)f (v∗)dωdv∗.
Q is a spatially local operator!A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 47: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/47.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Kernels
Maxwellian gas: B(|v − v∗|, cos θ) = const.
Hard Sphere gas: B(|v − v∗|, cos θ) = const ·|v − v∗|.Variable Hard Sphere (VHS) gas:B(|v − v∗|, cos θ) = const ·|v − v∗|α. (generalizes both casesabove)
Of course, one can cook up many more complicated collisionkernels.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 48: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/48.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Kernels
Maxwellian gas: B(|v − v∗|, cos θ) = const.
Hard Sphere gas: B(|v − v∗|, cos θ) = const ·|v − v∗|.
Variable Hard Sphere (VHS) gas:B(|v − v∗|, cos θ) = const ·|v − v∗|α. (generalizes both casesabove)
Of course, one can cook up many more complicated collisionkernels.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 49: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/49.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Kernels
Maxwellian gas: B(|v − v∗|, cos θ) = const.
Hard Sphere gas: B(|v − v∗|, cos θ) = const ·|v − v∗|.Variable Hard Sphere (VHS) gas:B(|v − v∗|, cos θ) = const ·|v − v∗|α. (generalizes both casesabove)
Of course, one can cook up many more complicated collisionkernels.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 50: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/50.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Collisions
Collision Kernels
Maxwellian gas: B(|v − v∗|, cos θ) = const.
Hard Sphere gas: B(|v − v∗|, cos θ) = const ·|v − v∗|.Variable Hard Sphere (VHS) gas:B(|v − v∗|, cos θ) = const ·|v − v∗|α. (generalizes both casesabove)
Of course, one can cook up many more complicated collisionkernels.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 51: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/51.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Summary
Challenges
Dimensionality: Eleven nested loops (naively) → horrible!
Timestepping: Different for transport and collision→ Splitting (next section)
Shock Waves (Transport → Section 3)
Collision operator: Deepest dimensionality. Can we do betterthan five nested loops at each point? (Collision → Section 4)
Positivity (if there is time)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 52: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/52.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Summary
Challenges
Dimensionality: Eleven nested loops (naively) → horrible!
Timestepping: Different for transport and collision→ Splitting (next section)
Shock Waves (Transport → Section 3)
Collision operator: Deepest dimensionality. Can we do betterthan five nested loops at each point? (Collision → Section 4)
Positivity (if there is time)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 53: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/53.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Summary
Challenges
Dimensionality: Eleven nested loops (naively) → horrible!
Timestepping: Different for transport and collision→ Splitting (next section)
Shock Waves (Transport → Section 3)
Collision operator: Deepest dimensionality. Can we do betterthan five nested loops at each point? (Collision → Section 4)
Positivity (if there is time)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 54: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/54.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Summary
Challenges
Dimensionality: Eleven nested loops (naively) → horrible!
Timestepping: Different for transport and collision→ Splitting (next section)
Shock Waves (Transport → Section 3)
Collision operator: Deepest dimensionality. Can we do betterthan five nested loops at each point? (Collision → Section 4)
Positivity (if there is time)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 55: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/55.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Summary
Challenges
Dimensionality: Eleven nested loops (naively) → horrible!
Timestepping: Different for transport and collision→ Splitting (next section)
Shock Waves (Transport → Section 3)
Collision operator: Deepest dimensionality. Can we do betterthan five nested loops at each point? (Collision → Section 4)
Positivity (if there is time)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 56: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/56.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Summary
Questions?
?
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Outline
1 IntroductionThe EquationCollisions
2 Time-Step SplittingOperator splittingRunge-Kutta
3 TransportFlux-Balance MethodsSemi-Lagrangian Methods
4 Collision OperatorDiscretizationBobylev-Rjasanow’s Integral Transform MethodPareschi-Russo’s Spectral MethodMouhot-Pareschi’s Sub-N6 Method
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Intro
General Framework
Semi-discrete form of Boltzmann equation
dy
dt= f1(y) + f2(y)
f1 and f2 may have significantly different properties
f1 represents convectionf2 is usually a stiff discretization of collision
Possible nonlinearity of f1: high-order accurate explicit method
Highly stiff f2: implicit method
Conservation? Positivity?
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 59: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/59.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Intro
General Framework
Semi-discrete form of Boltzmann equation
dy
dt= f1(y) + f2(y)
f1 and f2 may have significantly different properties
f1 represents convectionf2 is usually a stiff discretization of collision
Possible nonlinearity of f1: high-order accurate explicit method
Highly stiff f2: implicit method
Conservation? Positivity?
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 60: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/60.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Intro
Splitting Methods
Strategy 1: operator splittingS∆t
1 and S∆t2 are solution operators for
dy
dt= f1(y)
dy
dt= f2(y), (1)
respectivelyAttempt to approximate S∆t with S∆t
1 and S∆t2 :
S∆t ≈ p(S∆t1 ,S∆t
2 )
Strategy 2: Runge-Kutta split methods
Create two separate Runge-Kutta methods for each of theODE’s in (1), but evolve them togetherHigh-order accuracy requires coupling conditions between thetwo methodsDisadvantage: requires semi-discrete form of PDE
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Intro
Splitting Methods
Strategy 1: operator splittingS∆t
1 and S∆t2 are solution operators for
dy
dt= f1(y)
dy
dt= f2(y), (1)
respectivelyAttempt to approximate S∆t with S∆t
1 and S∆t2 :
S∆t ≈ p(S∆t1 ,S∆t
2 )
Strategy 2: Runge-Kutta split methodsCreate two separate Runge-Kutta methods for each of theODE’s in (1), but evolve them togetherHigh-order accuracy requires coupling conditions between thetwo methodsDisadvantage: requires semi-discrete form of PDE
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 62: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/62.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Operator splitting
Operator Splitting (1/2)
First guess: sequential application
S∆t ≈ S∆t1 S∆t
2
First-order in timeVery simple to formulate and elementary to implement
“Strang splitting”
S∆t ≈ S∆t/21 S∆t
2 S∆t/21
Second-order in timeNot much more difficult than first-order splittingProbably the most popular operator splitting algorithm usedtoday
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 63: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/63.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Operator splitting
Operator Splitting (1/2)
First guess: sequential application
S∆t ≈ S∆t1 S∆t
2
First-order in timeVery simple to formulate and elementary to implement
“Strang splitting”
S∆t ≈ S∆t/21 S∆t
2 S∆t/21
Second-order in timeNot much more difficult than first-order splittingProbably the most popular operator splitting algorithm usedtoday
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 64: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/64.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Operator splitting
Operator Splitting (2/2)
Higher-order accuracy?
Methods exist, but suffer from stability problemsDia has developed 3rd, 4th order schemes
Conservation?
Ohwada: second-order scheme
S∆t ≈ 1
2
(I + S∆t
2 S∆t2
)S∆t
1
Preserves mass, momemtum, energy (assuming the discretecollision operator does as well)Slightly more work compared to Strang splitting
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 65: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/65.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Operator splitting
Operator Splitting (2/2)
Higher-order accuracy?
Methods exist, but suffer from stability problemsDia has developed 3rd, 4th order schemes
Conservation?
Ohwada: second-order scheme
S∆t ≈ 1
2
(I + S∆t
2 S∆t2
)S∆t
1
Preserves mass, momemtum, energy (assuming the discretecollision operator does as well)Slightly more work compared to Strang splitting
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 66: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/66.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Runge-Kutta
Runge-Kutta methods
“Usual” s-stage Runge-Kutta methods have the form
yn+1 = yn + ∆ts∑
j=1
bj f (tn + cj∆t,Yj)
Yi = yn + ∆ts∑
j=1
aij f (tn + cj∆t,Yj)
where aij is an s × s matrix and bj and cj are s × 1 matrices
Can define explicit and implicit RK methods in this way
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Runge-Kutta
More Runge-Kutta
Implicit-explicit (IMEX) RK idea: use different RK methodsfor each different term:
yn+1 =yn + ∆ts∑
j=1
bj f1(tn + cj∆t,Yj)+
∆ts∑
j=1
bj f2(tn + cj∆t,Yj),
Yi =yn + ∆ts∑
j=1
aij f1(tn + cj∆t,Yj)+
∆ts∑
j=1
aij f2(tn + cj∆t,Yj)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 68: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/68.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Runge-Kutta
IMEX methods
What kind of RK methods to use for each operator?
Use an implicit RK method for the collision (f2)use an explicit RK method for the transport (f1)
Some examples of IMEX methods
First-order Euler explicit-implicitSecond-order methods: ‘midpoint’ splitting, Strang splitting,Jin splitting, ...Many higher-order methods
IMEX methods for the Boltzmann equation
Require semi-discrete form of the PDEDesire positivityTVD/SSP property ‘in the hyperbolic limit’ would be nice....
Can’t have your cake and eat it, too
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 69: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/69.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Runge-Kutta
IMEX methods
What kind of RK methods to use for each operator?
Use an implicit RK method for the collision (f2)use an explicit RK method for the transport (f1)
Some examples of IMEX methods
First-order Euler explicit-implicitSecond-order methods: ‘midpoint’ splitting, Strang splitting,Jin splitting, ...Many higher-order methods
IMEX methods for the Boltzmann equation
Require semi-discrete form of the PDEDesire positivityTVD/SSP property ‘in the hyperbolic limit’ would be nice....
Can’t have your cake and eat it, too
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 70: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/70.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Runge-Kutta
IMEX methods
What kind of RK methods to use for each operator?
Use an implicit RK method for the collision (f2)use an explicit RK method for the transport (f1)
Some examples of IMEX methods
First-order Euler explicit-implicitSecond-order methods: ‘midpoint’ splitting, Strang splitting,Jin splitting, ...Many higher-order methods
IMEX methods for the Boltzmann equation
Require semi-discrete form of the PDEDesire positivityTVD/SSP property ‘in the hyperbolic limit’ would be nice....
Can’t have your cake and eat it, too
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 71: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/71.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Runge-Kutta
IMEX methods
What kind of RK methods to use for each operator?
Use an implicit RK method for the collision (f2)use an explicit RK method for the transport (f1)
Some examples of IMEX methods
First-order Euler explicit-implicitSecond-order methods: ‘midpoint’ splitting, Strang splitting,Jin splitting, ...Many higher-order methods
IMEX methods for the Boltzmann equation
Require semi-discrete form of the PDEDesire positivityTVD/SSP property ‘in the hyperbolic limit’ would be nice....
Can’t have your cake and eat it, too
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 72: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/72.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Conclusion
What do people actually do?
Mostly operator splitting (Strang)
Splitting is, of course, not the only answer
Could use just about any ODE discretization you want
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 73: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/73.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Conclusion
What do people actually do?
Mostly operator splitting (Strang)
Splitting is, of course, not the only answer
Could use just about any ODE discretization you want
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 74: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/74.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Conclusion
Questions?
?
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 75: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/75.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Outline
1 IntroductionThe EquationCollisions
2 Time-Step SplittingOperator splittingRunge-Kutta
3 TransportFlux-Balance MethodsSemi-Lagrangian Methods
4 Collision OperatorDiscretizationBobylev-Rjasanow’s Integral Transform MethodPareschi-Russo’s Spectral MethodMouhot-Pareschi’s Sub-N6 Method
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Intro
The Vlasov Equation
Recall that the transport part of the Boltzmann equation isthe Vlasov equation:
∂f
∂t+ v · ∇x f + F · ∇v f = 0.
Nonlinear: F depends on fConservation law: v is x-independent and F is v -independent
Many methods exist to solve these types of equations
What we’ll talk about: methods designed specifically for theVlasov/Boltzmann equation
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Intro
The Vlasov Equation
Recall that the transport part of the Boltzmann equation isthe Vlasov equation:
∂f
∂t+ v · ∇x f + F · ∇v f = 0.
Nonlinear: F depends on fConservation law: v is x-independent and F is v -independent
Many methods exist to solve these types of equations
What we’ll talk about: methods designed specifically for theVlasov/Boltzmann equation
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 78: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/78.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Intro
Argh, More Splitting?
Complexity: six-dimensional transport equation → two3-dimensional systems is more tractable
∂f
∂t= v · ∇x f
∂f
∂t= F · ∇v f
Go even farther: split three dimensional problems into three1-dimensional problems?
Consider simple 1D conservation law:
∂f
∂t+∂ (vf )
∂x= 0
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Intro
Argh, More Splitting?
Complexity: six-dimensional transport equation → two3-dimensional systems is more tractable
∂f
∂t= v · ∇x f
∂f
∂t= F · ∇v f
Go even farther: split three dimensional problems into three1-dimensional problems?
Consider simple 1D conservation law:
∂f
∂t+∂ (vf )
∂x= 0
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Flux-Balance Methods
Flux-Balance Methods
Finite-volume degrees of freedom:
f ni =
∫Ii
f (x , t)dx
Method of characteristics: f is constant along x = X (t),where X (t) solves the ODE
dX (s)
dx= v(s,X (s))
X (tn+1) = x .
f (x , tn+1) = f (X (tn; tn+1, x), tn)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Flux-Balance Methods
Flux-Balance Methods
Finite-volume degrees of freedom:
f ni =
∫Ii
f (x , t)dx
Method of characteristics: f is constant along x = X (t),where X (t) solves the ODE
dX (s)
dx= v(s,X (s))
X (tn+1) = x .
f (x , tn+1) = f (X (tn; tn+1, x), tn)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Flux-Balance Methods
Method of Characteristics
X(tn; tn+1, x2)
x1 x2tn+1
tn
X(tn; tn+1, x1)
x
t
x1 x2
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Flux-Balance Methods
Flux-Balance Methods
Use characteristics for finite-volume:∫ xi+1/2
xi−1/2
f (tn+1, x)dx =
∫ X (tn,tn+1,xi+1/2)
X (tn;tn+1,xi−1/2)f (tn, x)dx
Define flux terms:
Φi+1/2(tn) =
∫ xi+1/2
xi+1/2−v∆tf (tn, x)dx
Rewrite first equation as∫ xi+1/2
xi−1/2
f (tn+1, x)dx =
∫ xi+1/2
xi−1/2
f (tn, x)dx+Φi−1/2(tn)− Φi+1/2(tn)
∆x
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Flux-Balance Methods
Flux-Balance Methods
Use characteristics for finite-volume:∫ xi+1/2
xi−1/2
f (tn+1, x)dx =
∫ X (tn,tn+1,xi+1/2)
X (tn;tn+1,xi−1/2)f (tn, x)dx
Define flux terms:
Φi+1/2(tn) =
∫ xi+1/2
xi+1/2−v∆tf (tn, x)dx
Rewrite first equation as∫ xi+1/2
xi−1/2
f (tn+1, x)dx =
∫ xi+1/2
xi−1/2
f (tn, x)dx+Φi−1/2(tn)− Φi+1/2(tn)
∆x
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Flux-Balance Methods
Flux-Balance Methods
Use characteristics for finite-volume:∫ xi+1/2
xi−1/2
f (tn+1, x)dx =
∫ X (tn,tn+1,xi+1/2)
X (tn;tn+1,xi−1/2)f (tn, x)dx
Define flux terms:
Φi+1/2(tn) =
∫ xi+1/2
xi+1/2−v∆tf (tn, x)dx
Rewrite first equation as∫ xi+1/2
xi−1/2
f (tn+1, x)dx =
∫ xi+1/2
xi−1/2
f (tn, x)dx+Φi−1/2(tn)− Φi+1/2(tn)
∆x
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Flux-Balance Methods
Flux-Balance Scheme
Leads to the finite-volume fully-discrete scheme
f n+1i = f n
i +Φi−1/2(tn)− Φi+1/2(tn)
∆x
How to evolve the scheme? Need to compute fluxes Φ.
No time-step restriction!Explicitly compute integral definition (reconstruct f )Compute pointwise primitive values (reconstruct primitive, F )
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Flux-Balance Methods
Flux-Balance Scheme
Leads to the finite-volume fully-discrete scheme
f n+1i = f n
i +Φi−1/2(tn)− Φi+1/2(tn)
∆x
How to evolve the scheme? Need to compute fluxes Φ.
No time-step restriction!Explicitly compute integral definition (reconstruct f )Compute pointwise primitive values (reconstruct primitive, F )
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Flux-Balance Methods
Flux-Balance – reconstructing f : PFC
The ‘Positive and Flux-Conservative’ (PFC) method
Third-order reconstruction using nonlinear slope limiters
f is reconstructed as a quadratic polynomial
fh(x) = fi + ε+p+2 (x)(fi+1 − fi ) + ε−p−2 (x)(fi − fi−1)
Preserves positivity of f
Reconstruction preserves cell averages (mass)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Flux-Balance Methods
Flux-Balance – reconstructing F : WENO
‘PWENO’: Pointwise WENO
Same principle as WENOReconstruction requires the primitive values inside cells (hencethe ‘pointwise’ appellation)More expensive than WENO: weights must be tailoreddepending on location → requires linear solveInherits spurious oscillation-reducing property of non-oscillatoryschemesCannot prove positivity/conservation
Use PWENO to reconstruct F (xj+1/2 − v∆t)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Flux-Balance Methods
Flux-Balance – reconstructing F : WENO
‘PWENO’: Pointwise WENO
Same principle as WENOReconstruction requires the primitive values inside cells (hencethe ‘pointwise’ appellation)More expensive than WENO: weights must be tailoreddepending on location → requires linear solveInherits spurious oscillation-reducing property of non-oscillatoryschemesCannot prove positivity/conservation
Use PWENO to reconstruct F (xj+1/2 − v∆t)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Semi-Lagrangian Methods
The Semi-Lagrangian Method
Very similar in spirit to flux-balance method: follow thecharacteristics
Degrees of freedom: point-values f n(xi )
Recall:
dX (s)
dx= v(s,X (s))
X (tn+1) = x
f (x , tn+1) = f (X (tn; tn+1, x), tn)
The semi-Lagrangian method:
f n+1(xi ) = f n(X (tn; tn+1, xi )
No time-step restriction!
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Semi-Lagrangian Methods
The Semi-Lagrangian Method
Very similar in spirit to flux-balance method: follow thecharacteristics
Degrees of freedom: point-values f n(xi )
Recall:
dX (s)
dx= v(s,X (s))
X (tn+1) = x
f (x , tn+1) = f (X (tn; tn+1, x), tn)
The semi-Lagrangian method:
f n+1(xi ) = f n(X (tn; tn+1, xi )
No time-step restriction!
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Semi-Lagrangian Methods
The Semi-Lagrangian Method
Very similar in spirit to flux-balance method: follow thecharacteristics
Degrees of freedom: point-values f n(xi )
Recall:
dX (s)
dx= v(s,X (s))
X (tn+1) = x
f (x , tn+1) = f (X (tn; tn+1, x), tn)
The semi-Lagrangian method:
f n+1(xi ) = f n(X (tn; tn+1, xi )
No time-step restriction!
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Semi-Lagrangian Methods
Semi-Lagrangian Schemes
We have f n(xi ) for all i
Need to interpolate to X (tn; tn+1, xi ) = xi − v∆t to advancethe scheme
Lagrange interpolations
Implementable to high-order interpolantsConservation: only with a central stencil, linearconstant-coefficient advection PDE
Hermite interpolations
Match function values and cell interface derivative valuesConservation: same as Lagrange interpolants
PWENO interpolations
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Semi-Lagrangian Methods
Semi-Lagrangian Schemes
We have f n(xi ) for all i
Need to interpolate to X (tn; tn+1, xi ) = xi − v∆t to advancethe scheme
Lagrange interpolations
Implementable to high-order interpolantsConservation: only with a central stencil, linearconstant-coefficient advection PDE
Hermite interpolations
Match function values and cell interface derivative valuesConservation: same as Lagrange interpolants
PWENO interpolations
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Semi-Lagrangian Methods
Semi-Lagrangian Schemes
We have f n(xi ) for all i
Need to interpolate to X (tn; tn+1, xi ) = xi − v∆t to advancethe scheme
Lagrange interpolations
Implementable to high-order interpolantsConservation: only with a central stencil, linearconstant-coefficient advection PDE
Hermite interpolations
Match function values and cell interface derivative valuesConservation: same as Lagrange interpolants
PWENO interpolations
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 97: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/97.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Semi-Lagrangian Methods
Semi-Lagrangian Schemes
We have f n(xi ) for all i
Need to interpolate to X (tn; tn+1, xi ) = xi − v∆t to advancethe scheme
Lagrange interpolations
Implementable to high-order interpolantsConservation: only with a central stencil, linearconstant-coefficient advection PDE
Hermite interpolations
Match function values and cell interface derivative valuesConservation: same as Lagrange interpolants
PWENO interpolations
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 98: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/98.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Others
Other Solvers
Spectral methods – computational savings: FFT
Finite-difference methods – Conservation: mass, energy
DG methods – positivity
Can’t have your cake and eat it, too
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Others
Other Solvers
Spectral methods – computational savings: FFT
Finite-difference methods – Conservation: mass, energy
DG methods – positivity
Can’t have your cake and eat it, too
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 100: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/100.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Others
Other Solvers
Spectral methods – computational savings: FFT
Finite-difference methods – Conservation: mass, energy
DG methods – positivity
Can’t have your cake and eat it, too
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 101: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/101.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Others
Other Solvers
Spectral methods – computational savings: FFT
Finite-difference methods – Conservation: mass, energy
DG methods – positivity
Can’t have your cake and eat it, too
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 102: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/102.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Others
Questions?
?
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Outline
1 IntroductionThe EquationCollisions
2 Time-Step SplittingOperator splittingRunge-Kutta
3 TransportFlux-Balance MethodsSemi-Lagrangian Methods
4 Collision OperatorDiscretizationBobylev-Rjasanow’s Integral Transform MethodPareschi-Russo’s Spectral MethodMouhot-Pareschi’s Sub-N6 Method
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Generalities
The Collision Operator
∂f
∂t=
1
knQ(f , f )
Q(f , f ) =
∫R3
∫S2
B[f (v ′)f (v ′∗)− f (v)f (v∗)
]dωdv∗
Properties:
Structure: Integrate over all post-collision velocities, then overall deflection vectors: 5 integrals at each (x , v).Q is local in x → neglect space dependence in this section.Conserves mass, momentum, energy:∫
R3
Q(f , f )
1v|v |2
dv = 0
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Generalities
The Collision Operator
∂f
∂t=
1
knQ(f , f )
Q(f , f ) =
∫R3
∫S2
B[f (v ′)f (v ′∗)− f (v)f (v∗)
]dωdv∗
Properties:
Structure: Integrate over all post-collision velocities, then overall deflection vectors: 5 integrals at each (x , v).
Q is local in x → neglect space dependence in this section.Conserves mass, momentum, energy:∫
R3
Q(f , f )
1v|v |2
dv = 0
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Generalities
The Collision Operator
∂f
∂t=
1
knQ(f , f )
Q(f , f ) =
∫R3
∫S2
B[f (v ′)f (v ′∗)− f (v)f (v∗)
]dωdv∗
Properties:
Structure: Integrate over all post-collision velocities, then overall deflection vectors: 5 integrals at each (x , v).Q is local in x → neglect space dependence in this section.
Conserves mass, momentum, energy:∫R3
Q(f , f )
1v|v |2
dv = 0
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Generalities
The Collision Operator
∂f
∂t=
1
knQ(f , f )
Q(f , f ) =
∫R3
∫S2
B[f (v ′)f (v ′∗)− f (v)f (v∗)
]dωdv∗
Properties:
Structure: Integrate over all post-collision velocities, then overall deflection vectors: 5 integrals at each (x , v).Q is local in x → neglect space dependence in this section.Conserves mass, momentum, energy:∫
R3
Q(f , f )
1v|v |2
dv = 0
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Generalities
The Collision Operator–Steady-State
Evolves towards Gaussian in v Maxwellian
f∞(v) :=ρ
(2πT )d/2exp
(−|V − v |2
2T
)
Steady-state: Q(f∞, f∞) = 0
Does so even for full Boltzmann equation.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Generalities
The Collision Operator–Steady-State
Evolves towards Gaussian in v Maxwellian
f∞(v) :=ρ
(2πT )d/2exp
(−|V − v |2
2T
)
Steady-state: Q(f∞, f∞) = 0
Does so even for full Boltzmann equation.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 110: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/110.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Generalities
The Collision Operator–Steady-State
f∞(v) :=ρ
(2πT )d/2exp
(−|V − v |2
2T
).
Density:
ρ :=
∫Rd
f (v)dv
Mean/Bulk Velocity:
V :=1
ρ
∫Rd
vf (v)dv
Temperature:
T :=1
3ρ
∫Rd
|V − v |2f (v)dv
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Generalities
The Collision Operator–Steady-State
f∞(v) :=ρ
(2πT )d/2exp
(−|V − v |2
2T
).
Density:
ρ :=
∫Rd
f (v)dv
Mean/Bulk Velocity:
V :=1
ρ
∫Rd
vf (v)dv
Temperature:
T :=1
3ρ
∫Rd
|V − v |2f (v)dv
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Generalities
The Collision Operator–Steady-State
f∞(v) :=ρ
(2πT )d/2exp
(−|V − v |2
2T
).
Density:
ρ :=
∫Rd
f (v)dv
Mean/Bulk Velocity:
V :=1
ρ
∫Rd
vf (v)dv
Temperature:
T :=1
3ρ
∫Rd
|V − v |2f (v)dv
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Generalities
Simplifying the Collision Operator
Q(f , f ) =
∫R3
∫S2
B[f (v ′)f (v ′∗)− f (v)f (v∗)
]dωdv∗
Linearize: f := f∞ + g
Q lin(f , f ) ≈∫R3
∫S2
B[g(v ′)f∞(v ′∗) + f∞(v ′)g(v ′∗)
− g(v)f∞(v∗)− f∞(v)g(v∗)]dωdv∗
∫R3
k(v , v ′)f∞(v)f (v ′)− k(v ′, v)f∞(v ′)f (v)dv ′
→Linear Vlasov-Boltzmann Equation
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 114: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/114.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Generalities
Simplifying the Collision Operator
Q(f , f ) =
∫R3
∫S2
B[f (v ′)f (v ′∗)− f (v)f (v∗)
]dωdv∗
Linearize: f := f∞ + g
Q lin(f , f ) ≈∫R3
∫S2
B[g(v ′)f∞(v ′∗) + f∞(v ′)g(v ′∗)
− g(v)f∞(v∗)− f∞(v)g(v∗)]dωdv∗
∫R3
k(v , v ′)f∞(v)f (v ′)− k(v ′, v)f∞(v ′)f (v)dv ′
→Linear Vlasov-Boltzmann Equation
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 115: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/115.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Generalities
Simplifying the Collision Operator
Q(f , f ) =
∫R3
∫S2
B[f (v ′)f (v ′∗)− f (v)f (v∗)
]dωdv∗
Linearize: f := f∞ + g
Q lin(f , f ) ≈∫R3
∫S2
B[g(v ′)f∞(v ′∗) + f∞(v ′)g(v ′∗)
− g(v)f∞(v∗)− f∞(v)g(v∗)]dωdv∗
∫R3
k(v , v ′)f∞(v)f (v ′)− k(v ′, v)f∞(v ′)f (v)dv ′
→Linear Vlasov-Boltzmann Equation
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 116: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/116.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Generalities
Simplifying the Collision Operator
Q(f , f ) =
∫R3
∫S2
B[f (v ′)f (v ′∗)− f (v)f (v∗)
]dωdv∗
Linearize: f := f∞ + g
Q lin(f , f ) ≈∫R3
∫S2
B[g(v ′)f∞(v ′∗) + f∞(v ′)g(v ′∗)
− g(v)f∞(v∗)− f∞(v)g(v∗)]dωdv∗
∫R3
k(v , v ′)f∞(v)f (v ′)− k(v ′, v)f∞(v ′)f (v)dv ′
→Linear Vlasov-Boltzmann Equation
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 117: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/117.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Generalities
Simplifying the Collision Operator
Q(f , f ) =
∫R3
∫S2
B[f (v ′)f (v ′∗)− f (v)f (v∗)
]dωdv∗
Linearize: f := f∞ + g
Q lin(f , f ) ≈∫R3
∫S2
B[g(v ′)f∞(v ′∗) + f∞(v ′)g(v ′∗)
− g(v)f∞(v∗)− f∞(v)g(v∗)]dωdv∗
∫R3
k(v , v ′)f∞(v)f (v ′)− k(v ′, v)f∞(v ′)f (v)dv ′
→Linear Vlasov-Boltzmann Equation
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 118: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/118.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Generalities
Simplifying the Collision Operator
Q(f , f ) =
∫R3
∫S2
B[f (v ′)f (v ′∗)− f (v)f (v∗)
]dωdv∗
Even simpler: Bhatnagar-Gross-Krook (BGK) Approximation
QBGK(f ) =1
τ(f∞ − f )
Still stronger than (i.e. implies) Navier-Stokes.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Generalities
Simplifying the Collision Operator
Q(f , f ) =
∫R3
∫S2
B[f (v ′)f (v ′∗)− f (v)f (v∗)
]dωdv∗
Even simpler: Bhatnagar-Gross-Krook (BGK) Approximation
QBGK(f ) =1
τ(f∞ − f )
Still stronger than (i.e. implies) Navier-Stokes.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 120: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/120.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Generalities
Simplifying the Collision Operator
Q(f , f ) =
∫R3
∫S2
B[f (v ′)f (v ′∗)− f (v)f (v∗)
]dωdv∗
Even simpler: Bhatnagar-Gross-Krook (BGK) Approximation
QBGK(f ) =1
τ(f∞ − f )
Still stronger than (i.e. implies) Navier-Stokes.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 121: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/121.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Discretizing Velocity Space
v -space: infinite.
Computers: finite. →Something’s got to go.
Lemma (Compact support for Q)
Suppose suppv (f ) ⊂ B(0,R). Then
1 suppv (Q(f , f )(v)) ⊂ B(0,R√
2).
2 Change of variables g := v − v∗. We can restrictg ∈ B(0, 2R) with no change:
Q =
∫B(2R)×S2
B(|g |, θ)[f ′(v ′)f (v ′∗)− f (v)f (v − g)
]dωdg
3 With this limitation, we automatically also havev∗ = v − g , v ′, v ′∗ ∈ B(0, (2 +
√2)R).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Discretization
Discretizing Velocity Space
v -space: infinite. Computers: finite.
→Something’s got to go.
Lemma (Compact support for Q)
Suppose suppv (f ) ⊂ B(0,R). Then
1 suppv (Q(f , f )(v)) ⊂ B(0,R√
2).
2 Change of variables g := v − v∗. We can restrictg ∈ B(0, 2R) with no change:
Q =
∫B(2R)×S2
B(|g |, θ)[f ′(v ′)f (v ′∗)− f (v)f (v − g)
]dωdg
3 With this limitation, we automatically also havev∗ = v − g , v ′, v ′∗ ∈ B(0, (2 +
√2)R).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 123: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/123.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Discretizing Velocity Space
v -space: infinite. Computers: finite. →Something’s got to go.
Lemma (Compact support for Q)
Suppose suppv (f ) ⊂ B(0,R). Then
1 suppv (Q(f , f )(v)) ⊂ B(0,R√
2).
2 Change of variables g := v − v∗. We can restrictg ∈ B(0, 2R) with no change:
Q =
∫B(2R)×S2
B(|g |, θ)[f ′(v ′)f (v ′∗)− f (v)f (v − g)
]dωdg
3 With this limitation, we automatically also havev∗ = v − g , v ′, v ′∗ ∈ B(0, (2 +
√2)R).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 124: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/124.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Discretizing Velocity Space
v -space: infinite. Computers: finite. →Something’s got to go.
Lemma (Compact support for Q)
Suppose suppv (f ) ⊂ B(0,R). Then
1 suppv (Q(f , f )(v)) ⊂ B(0,R√
2).
2 Change of variables g := v − v∗. We can restrictg ∈ B(0, 2R) with no change:
Q =
∫B(2R)×S2
B(|g |, θ)[f ′(v ′)f (v ′∗)− f (v)f (v − g)
]dωdg
3 With this limitation, we automatically also havev∗ = v − g , v ′, v ′∗ ∈ B(0, (2 +
√2)R).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 125: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/125.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Discretizing Velocity Space
v -space: infinite. Computers: finite. →Something’s got to go.
Lemma (Compact support for Q)
Suppose suppv (f ) ⊂ B(0,R). Then
1 suppv (Q(f , f )(v)) ⊂ B(0,R√
2).
2 Change of variables g := v − v∗. We can restrictg ∈ B(0, 2R) with no change:
Q =
∫B(2R)×S2
B(|g |, θ)[f ′(v ′)f (v ′∗)− f (v)f (v − g)
]dωdg
3 With this limitation, we automatically also havev∗ = v − g , v ′, v ′∗ ∈ B(0, (2 +
√2)R).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 126: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/126.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Discretizing Velocity Space
v -space: infinite. Computers: finite. →Something’s got to go.
Lemma (Compact support for Q)
Suppose suppv (f ) ⊂ B(0,R). Then
1 suppv (Q(f , f )(v)) ⊂ B(0,R√
2).
2 Change of variables g := v − v∗. We can restrictg ∈ B(0, 2R) with no change:
Q =
∫B(2R)×S2
B(|g |, θ)[f ′(v ′)f (v ′∗)− f (v)f (v − g)
]dωdg
3 With this limitation, we automatically also havev∗ = v − g , v ′, v ′∗ ∈ B(0, (2 +
√2)R).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 127: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/127.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Discretizing Velocity Space
v -space: infinite. Computers: finite. →Something’s got to go.
Lemma (Compact support for Q)
Suppose suppv (f ) ⊂ B(0,R). Then
1 suppv (Q(f , f )(v)) ⊂ B(0,R√
2).
2 Change of variables g := v − v∗. We can restrictg ∈ B(0, 2R) with no change:
Q =
∫B(2R)×S2
B(|g |, θ)[f ′(v ′)f (v ′∗)− f (v)f (v − g)
]dωdg
3 With this limitation, we automatically also havev∗ = v − g , v ′, v ′∗ ∈ B(0, (2 +
√2)R).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 128: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/128.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Compact support for Q
velocity bound
√2R (2 +
√2)R
suppv Q(f, f)
Rsuppv f
suppv (f ) grows by a factor of√
2 each timestep—for any ∆t > 0!
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Discretization
Compact support for Q
velocity bound
√2R (2 +
√2)R
suppv Q(f, f)
Rsuppv f
suppv (f ) grows by a factor of√
2 each timestep
—for any ∆t > 0!
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 130: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/130.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Compact support for Q
velocity bound
√2R (2 +
√2)R
suppv Q(f, f)
Rsuppv f
suppv (f ) grows by a factor of√
2 each timestep—for any ∆t > 0!A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transforms in v -Space: Why?
Despite negative result:
Keep fixed, bounded discretization in v -space.Hope for the best. (i.e. that f is small where we truncate)
Need to overcome N8 complexity.Strategy: Look for convolution structure, carry out using FFT.
Convolution carried out directly: O(N2)
Convolution carried out using FFT: O(N log N)
Hope: Similar savings!→Investigate Fourier Transforms in v -Space.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 132: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/132.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transforms in v -Space: Why?
Despite negative result:Keep fixed, bounded discretization in v -space.Hope for the best.
(i.e. that f is small where we truncate)
Need to overcome N8 complexity.Strategy: Look for convolution structure, carry out using FFT.
Convolution carried out directly: O(N2)
Convolution carried out using FFT: O(N log N)
Hope: Similar savings!→Investigate Fourier Transforms in v -Space.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 133: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/133.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transforms in v -Space: Why?
Despite negative result:Keep fixed, bounded discretization in v -space.Hope for the best. (i.e. that f is small where we truncate)
Need to overcome N8 complexity.Strategy: Look for convolution structure, carry out using FFT.
Convolution carried out directly: O(N2)
Convolution carried out using FFT: O(N log N)
Hope: Similar savings!→Investigate Fourier Transforms in v -Space.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 134: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/134.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transforms in v -Space: Why?
Despite negative result:Keep fixed, bounded discretization in v -space.Hope for the best. (i.e. that f is small where we truncate)
Need to overcome N8 complexity.
Strategy: Look for convolution structure, carry out using FFT.
Convolution carried out directly: O(N2)
Convolution carried out using FFT: O(N log N)
Hope: Similar savings!→Investigate Fourier Transforms in v -Space.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 135: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/135.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transforms in v -Space: Why?
Despite negative result:Keep fixed, bounded discretization in v -space.Hope for the best. (i.e. that f is small where we truncate)
Need to overcome N8 complexity.Strategy: Look for convolution structure,
carry out using FFT.
Convolution carried out directly: O(N2)
Convolution carried out using FFT: O(N log N)
Hope: Similar savings!→Investigate Fourier Transforms in v -Space.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 136: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/136.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transforms in v -Space: Why?
Despite negative result:Keep fixed, bounded discretization in v -space.Hope for the best. (i.e. that f is small where we truncate)
Need to overcome N8 complexity.Strategy: Look for convolution structure, carry out using FFT.
Convolution carried out directly: O(N2)
Convolution carried out using FFT: O(N log N)
Hope: Similar savings!→Investigate Fourier Transforms in v -Space.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 137: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/137.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transforms in v -Space: Why?
Despite negative result:Keep fixed, bounded discretization in v -space.Hope for the best. (i.e. that f is small where we truncate)
Need to overcome N8 complexity.Strategy: Look for convolution structure, carry out using FFT.
Convolution carried out directly: O(N2)
Convolution carried out using FFT: O(N log N)
Hope: Similar savings!→Investigate Fourier Transforms in v -Space.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 138: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/138.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transforms in v -Space: Why?
Despite negative result:Keep fixed, bounded discretization in v -space.Hope for the best. (i.e. that f is small where we truncate)
Need to overcome N8 complexity.Strategy: Look for convolution structure, carry out using FFT.
Convolution carried out directly: O(N2)
Convolution carried out using FFT: O(N log N)
Hope: Similar savings!→Investigate Fourier Transforms in v -Space.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 139: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/139.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transforms in v -Space: Why?
Despite negative result:Keep fixed, bounded discretization in v -space.Hope for the best. (i.e. that f is small where we truncate)
Need to overcome N8 complexity.Strategy: Look for convolution structure, carry out using FFT.
Convolution carried out directly: O(N2)
Convolution carried out using FFT: O(N log N)
Hope: Similar savings!
→Investigate Fourier Transforms in v -Space.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 140: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/140.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transforms in v -Space: Why?
Despite negative result:Keep fixed, bounded discretization in v -space.Hope for the best. (i.e. that f is small where we truncate)
Need to overcome N8 complexity.Strategy: Look for convolution structure, carry out using FFT.
Convolution carried out directly: O(N2)
Convolution carried out using FFT: O(N log N)
Hope: Similar savings!→Investigate Fourier Transforms in v -Space.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 141: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/141.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transform in v -Space
Introduce a lattice:
vk := hvk , hv > 0, k ∈ Z3
Approximate the Fourier Integral
ϕ(ξ) =
∫R3
f (v)e i(v ·ξ)dv
using the midpoint rule
ϕ(ξ) := h3v
∑k∈Z3
f (vk)e i(v ·ξ).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transform in v -Space
Introduce a lattice:
vk := hvk , hv > 0, k ∈ Z3
Approximate the Fourier Integral
ϕ(ξ) =
∫R3
f (v)e i(v ·ξ)dv
using the midpoint rule
ϕ(ξ) := h3v
∑k∈Z3
f (vk)e i(v ·ξ).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 143: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/143.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transform in v -Space
Introduce a lattice:
vk := hvk , hv > 0, k ∈ Z3
Approximate the Fourier Integral
ϕ(ξ) =
∫R3
f (v)e i(v ·ξ)dv
using the midpoint rule
ϕ(ξ) := h3v
∑k∈Z3
f (vk)e i(v ·ξ).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 144: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/144.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transform in v -Space
ϕ(ξ) := h3v
∑k∈Z3
f (vk)e i(v ·ξ).
But for which ξ do we need to evaluate this?
Pick a grid in ξ-space, too: ξj := hξj with hξ > 0 and j ∈ Z3.FFT requires: hvhξ = 2π/n for n ∈ N.Get n-periodicity in k and j :
ϕ(ξj) =∑k∈Z3
f (vk)e i 2πn
(k·j)
=∑k∈Z3
f (vk)e i 2πn
((k+ein)·j) =∑k∈Z3
f (vk)e i 2πn
(k·(j+ein))
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 145: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/145.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transform in v -Space
ϕ(ξ) := h3v
∑k∈Z3
f (vk)e i(v ·ξ).
But for which ξ do we need to evaluate this?Pick a grid in ξ-space, too:
ξj := hξj with hξ > 0 and j ∈ Z3.FFT requires: hvhξ = 2π/n for n ∈ N.Get n-periodicity in k and j :
ϕ(ξj) =∑k∈Z3
f (vk)e i 2πn
(k·j)
=∑k∈Z3
f (vk)e i 2πn
((k+ein)·j) =∑k∈Z3
f (vk)e i 2πn
(k·(j+ein))
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 146: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/146.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transform in v -Space
ϕ(ξ) := h3v
∑k∈Z3
f (vk)e i(v ·ξ).
But for which ξ do we need to evaluate this?Pick a grid in ξ-space, too: ξj := hξj
with hξ > 0 and j ∈ Z3.FFT requires: hvhξ = 2π/n for n ∈ N.Get n-periodicity in k and j :
ϕ(ξj) =∑k∈Z3
f (vk)e i 2πn
(k·j)
=∑k∈Z3
f (vk)e i 2πn
((k+ein)·j) =∑k∈Z3
f (vk)e i 2πn
(k·(j+ein))
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 147: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/147.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transform in v -Space
ϕ(ξ) := h3v
∑k∈Z3
f (vk)e i(v ·ξ).
But for which ξ do we need to evaluate this?Pick a grid in ξ-space, too: ξj := hξj with hξ > 0 and j ∈ Z3.
FFT requires: hvhξ = 2π/n for n ∈ N.Get n-periodicity in k and j :
ϕ(ξj) =∑k∈Z3
f (vk)e i 2πn
(k·j)
=∑k∈Z3
f (vk)e i 2πn
((k+ein)·j) =∑k∈Z3
f (vk)e i 2πn
(k·(j+ein))
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 148: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/148.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transform in v -Space
ϕ(ξ) := h3v
∑k∈Z3
f (vk)e i(v ·ξ).
But for which ξ do we need to evaluate this?Pick a grid in ξ-space, too: ξj := hξj with hξ > 0 and j ∈ Z3.FFT requires: hvhξ = 2π/n for n ∈ N.
Get n-periodicity in k and j :
ϕ(ξj) =∑k∈Z3
f (vk)e i 2πn
(k·j)
=∑k∈Z3
f (vk)e i 2πn
((k+ein)·j) =∑k∈Z3
f (vk)e i 2πn
(k·(j+ein))
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 149: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/149.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transform in v -Space
ϕ(ξ) := h3v
∑k∈Z3
f (vk)e i(v ·ξ).
But for which ξ do we need to evaluate this?Pick a grid in ξ-space, too: ξj := hξj with hξ > 0 and j ∈ Z3.FFT requires: hvhξ = 2π/n for n ∈ N.Get n-periodicity in k and j :
ϕ(ξj) =∑k∈Z3
f (vk)e i 2πn
(k·j)
=∑k∈Z3
f (vk)e i 2πn
((k+ein)·j) =∑k∈Z3
f (vk)e i 2πn
(k·(j+ein))
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 150: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/150.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transform in v -Space
ϕ(ξ) := h3v
∑k∈Z3
f (vk)e i(v ·ξ).
But for which ξ do we need to evaluate this?Pick a grid in ξ-space, too: ξj := hξj with hξ > 0 and j ∈ Z3.FFT requires: hvhξ = 2π/n for n ∈ N.Get n-periodicity in k and j :
ϕ(ξj) =∑k∈Z3
f (vk)e i 2πn
(k·j)
=∑k∈Z3
f (vk)e i 2πn
((k+ein)·j) =∑k∈Z3
f (vk)e i 2πn
(k·(j+ein))
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 151: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/151.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transform in v -Space
ϕ(ξ) := h3v
∑k∈Z3
f (vk)e i(v ·ξ).
But for which ξ do we need to evaluate this?Pick a grid in ξ-space, too: ξj := hξj with hξ > 0 and j ∈ Z3.FFT requires: hvhξ = 2π/n for n ∈ N.Get n-periodicity in k and j :
ϕ(ξj) =∑k∈Z3
f (vk)e i 2πn
(k·j)
=∑k∈Z3
f (vk)e i 2πn
((k+ein)·j)
=∑k∈Z3
f (vk)e i 2πn
(k·(j+ein))
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 152: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/152.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Fourier Transform in v -Space
ϕ(ξ) := h3v
∑k∈Z3
f (vk)e i(v ·ξ).
But for which ξ do we need to evaluate this?Pick a grid in ξ-space, too: ξj := hξj with hξ > 0 and j ∈ Z3.FFT requires: hvhξ = 2π/n for n ∈ N.Get n-periodicity in k and j :
ϕ(ξj) =∑k∈Z3
f (vk)e i 2πn
(k·j)
=∑k∈Z3
f (vk)e i 2πn
((k+ein)·j) =∑k∈Z3
f (vk)e i 2πn
(k·(j+ein))
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 153: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/153.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Faking Periodicity
So v -space becomes (artificially) 2L-periodic, with L = nhv/2.
R0
√2R0
R0suppv(f)
2L
suppv Q(f, f)
2R0
Avoid aliasing: choose L > 12 (√
2R0 + 2R0 + R0) = 3+√
22 R0.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 154: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/154.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Faking Periodicity
So v -space becomes (artificially) 2L-periodic, with L = nhv/2.
R0
√2R0
R0suppv(f)
2L
suppv Q(f, f)
2R0
Avoid aliasing: choose L > 12 (√
2R0 + 2R0 + R0) = 3+√
22 R0.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 155: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/155.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Discretization
Faking Periodicity
So v -space becomes (artificially) 2L-periodic, with L = nhv/2.
R0
√2R0
R0suppv(f)
2L
suppv Q(f, f)
2R0
Avoid aliasing: choose L > 12 (√
2R0 + 2R0 + R0) = 3+√
22 R0.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 156: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/156.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Overview
Overview
M H V O Order Ops Ms Mm En
DSMC � � � � N−1/2 N3 � � �IbrR � � � − N−2 N6 � ◦ ◦
gen. − − − � N−2 N6 log N � ◦ ◦BobR − 3 − − N−2 N6 log N � ◦ ◦ParRu � � � − N−∞ N6 � ◦ ◦MouP 2 3 − − N−∞ N5 log N � ◦ ◦
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 157: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/157.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Bobylev-Rjasanow’s Integral Transform Method
Idea:
Rewrite Q(f , f ) into convolution integrals.Exploit the Fast Fourier Transform to evaluate these integralsquickly.
skip the details
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Bobylev-Rjasanow’s Method
Lemma (Symmetric Representation)
The collision operator can be represented as
Q(f , f ) = 4
∫R3
∫R3
B(y , z)δ(z · y)[f (v + z)(f (v + y)
− f (v)f (v + y + z)]dydz ,
with
B(y , z) :=1
|y + z |B(|y + z |, y · (y + z)
|y ||y + z |).
skip the rest of this method
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Bobylev-Rjasanow’s Method
Theorem (Integral Transform Representation)
The collision operator for hard spheres can also be represented as
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2,
(note the implicit v-dependency)
where
Φ(v , ω) :=
∫ ∞−∞|ρ|f (v + ρω)dρ
Ψ(v , ω1, ω2) :=
∫ ∞−∞
∫ ∞∞|ρ1||ρ2|f (v + ρ1ω1 + ρ2ω2)dρ1dρ2.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 160: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/160.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Bobylev-Rjasanow’s Method
Theorem (Integral Transform Representation)
The collision operator for hard spheres can also be represented as
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2,
(note the implicit v-dependency) where
Φ(v , ω) :=
∫ ∞−∞|ρ|f (v + ρω)dρ
Ψ(v , ω1, ω2) :=
∫ ∞−∞
∫ ∞∞|ρ1||ρ2|f (v + ρ1ω1 + ρ2ω2)dρ1dρ2.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 161: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/161.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Bobylev-Rjasanow’s Method
Theorem (Integral Transform Representation)
The collision operator for hard spheres can also be represented as
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2,
(note the implicit v-dependency) where
Φ(v , ω) :=
∫ ∞−∞|ρ|f (v + ρω)dρ
Ψ(v , ω1, ω2) :=
∫ ∞−∞
∫ ∞∞|ρ1||ρ2|f (v + ρ1ω1 + ρ2ω2)dρ1dρ2.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 162: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/162.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Remarks on Bobylev-Rjasanow
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2,
We’ve grown an extra integral. (but also a δ-function)
Integrating over all pairs of orthogonal unit vectors:Apply cleverness to exploit structure.
Φ: “Generalized X-ray Transform”
Ψ: “Generalized Radon Transform”
Φ and Ψ of convolution type: good for FFT.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 163: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/163.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Remarks on Bobylev-Rjasanow
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2,
We’ve grown an extra integral. (but also a δ-function)
Integrating over all pairs of orthogonal unit vectors:Apply cleverness to exploit structure.
Φ: “Generalized X-ray Transform”
Ψ: “Generalized Radon Transform”
Φ and Ψ of convolution type: good for FFT.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 164: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/164.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Remarks on Bobylev-Rjasanow
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2,
We’ve grown an extra integral. (but also a δ-function)
Integrating over all pairs of orthogonal unit vectors:Apply cleverness to exploit structure.
Φ: “Generalized X-ray Transform”
Ψ: “Generalized Radon Transform”
Φ and Ψ of convolution type: good for FFT.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 165: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/165.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Remarks on Bobylev-Rjasanow
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2,
We’ve grown an extra integral. (but also a δ-function)
Integrating over all pairs of orthogonal unit vectors:Apply cleverness to exploit structure.
Φ: “Generalized X-ray Transform”
Ψ: “Generalized Radon Transform”
Φ and Ψ of convolution type: good for FFT.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 166: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/166.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Remarks on Bobylev-Rjasanow
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2,
We’ve grown an extra integral. (but also a δ-function)
Integrating over all pairs of orthogonal unit vectors:Apply cleverness to exploit structure.
Φ: “Generalized X-ray Transform”
Ψ: “Generalized Radon Transform”
Φ and Ψ of convolution type: good for FFT.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 167: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/167.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Rewriting X-Ray Transform→Fourier Transform
Φ(v , ω) = F−1[F [Φ](ξ, ω)](v , ω)
= F−1
[∫R3
Φ(v , ω)e i(v ·ξ)dv
](v , ω)
= F−1
[∫R3
∫ ∞−∞|ρ|f (v + ρω)dρe i(v ·ξ)dv
](v , ω)
(v := v + ρω) = F−1
[∫R3
∫ ∞−∞|ρ|f (v)e i((v−ρω)·ξ)dρdv
](v , ω)
= F−1
[ ∫R3
f (v)e i(v ·ξ)dv
∫ ∞−∞|ρ|e−i(ρω·ξ)dρ︸ ︷︷ ︸
d(ω·ξ):=F [|ρ|](ω·ξ):=
](v , ω)
= F−1 [ϕ(ξ)d(ξ · ω)] (v , ω)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 168: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/168.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Rewriting X-Ray Transform→Fourier Transform
Φ(v , ω) = F−1[F [Φ](ξ, ω)](v , ω)
= F−1
[∫R3
Φ(v , ω)e i(v ·ξ)dv
](v , ω)
= F−1
[∫R3
∫ ∞−∞|ρ|f (v + ρω)dρe i(v ·ξ)dv
](v , ω)
(v := v + ρω) = F−1
[∫R3
∫ ∞−∞|ρ|f (v)e i((v−ρω)·ξ)dρdv
](v , ω)
= F−1
[ ∫R3
f (v)e i(v ·ξ)dv
∫ ∞−∞|ρ|e−i(ρω·ξ)dρ︸ ︷︷ ︸
d(ω·ξ):=F [|ρ|](ω·ξ):=
](v , ω)
= F−1 [ϕ(ξ)d(ξ · ω)] (v , ω)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 169: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/169.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Rewriting X-Ray Transform→Fourier Transform
Φ(v , ω) = F−1[F [Φ](ξ, ω)](v , ω)
= F−1
[∫R3
Φ(v , ω)e i(v ·ξ)dv
](v , ω)
= F−1
[∫R3
∫ ∞−∞|ρ|f (v + ρω)dρe i(v ·ξ)dv
](v , ω)
(v := v + ρω) = F−1
[∫R3
∫ ∞−∞|ρ|f (v)e i((v−ρω)·ξ)dρdv
](v , ω)
= F−1
[ ∫R3
f (v)e i(v ·ξ)dv
∫ ∞−∞|ρ|e−i(ρω·ξ)dρ︸ ︷︷ ︸
d(ω·ξ):=F [|ρ|](ω·ξ):=
](v , ω)
= F−1 [ϕ(ξ)d(ξ · ω)] (v , ω)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 170: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/170.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Rewriting X-Ray Transform→Fourier Transform
Φ(v , ω) = F−1[F [Φ](ξ, ω)](v , ω)
= F−1
[∫R3
Φ(v , ω)e i(v ·ξ)dv
](v , ω)
= F−1
[∫R3
∫ ∞−∞|ρ|f (v + ρω)dρe i(v ·ξ)dv
](v , ω)
(v := v + ρω) = F−1
[∫R3
∫ ∞−∞|ρ|f (v)e i((v−ρω)·ξ)dρdv
](v , ω)
= F−1
[ ∫R3
f (v)e i(v ·ξ)dv
∫ ∞−∞|ρ|e−i(ρω·ξ)dρ︸ ︷︷ ︸
d(ω·ξ):=F [|ρ|](ω·ξ):=
](v , ω)
= F−1 [ϕ(ξ)d(ξ · ω)] (v , ω)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 171: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/171.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Rewriting X-Ray Transform→Fourier Transform
Φ(v , ω) = F−1[F [Φ](ξ, ω)](v , ω)
= F−1
[∫R3
Φ(v , ω)e i(v ·ξ)dv
](v , ω)
= F−1
[∫R3
∫ ∞−∞|ρ|f (v + ρω)dρe i(v ·ξ)dv
](v , ω)
(v := v + ρω) = F−1
[∫R3
∫ ∞−∞|ρ|f (v)e i((v−ρω)·ξ)dρdv
](v , ω)
= F−1
[ ∫R3
f (v)e i(v ·ξ)dv
∫ ∞−∞|ρ|e−i(ρω·ξ)dρ︸ ︷︷ ︸
d(ω·ξ):=F [|ρ|](ω·ξ):=
](v , ω)
= F−1 [ϕ(ξ)d(ξ · ω)] (v , ω)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
Rewriting X-Ray Transform→Fourier Transform
Φ(v , ω) = F−1[F [Φ](ξ, ω)](v , ω)
= F−1
[∫R3
Φ(v , ω)e i(v ·ξ)dv
](v , ω)
= F−1
[∫R3
∫ ∞−∞|ρ|f (v + ρω)dρe i(v ·ξ)dv
](v , ω)
(v := v + ρω) = F−1
[∫R3
∫ ∞−∞|ρ|f (v)e i((v−ρω)·ξ)dρdv
](v , ω)
= F−1
[ ∫R3
f (v)e i(v ·ξ)dv
∫ ∞−∞|ρ|e−i(ρω·ξ)dρ︸ ︷︷ ︸
d(ω·ξ):=F [|ρ|](ω·ξ):=
](v , ω)
= F−1 [ϕ(ξ)d(ξ · ω)] (v , ω)A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
The Algorithm
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2
Φ(v , ω) = F−1 [ϕ(ξ)d(ξ · ω)] (v , ω)]
Ψ(v , ω) = F−1[ϕ(ξ)d(ξ · ω1)d(ξ · ω2)](v , ω1, ω2)]
1 Fourier transform f → ϕ
2 For each node ω1 of a quadrature rule on S2:
1 For each node ω2 of a quadrature rule on ω⊥1 :
1 Compute FΦ(ξ) := ϕ(ξ)d(ξ · ω).2 Compute FΨ(ξ) := ϕ(ξ)d(ξ · ω1)d(ξ · ω2).3 Compute the inverse Fourier transform of FΦ and FΨ.
3 Add up contributions, giving Q(f , f ) for all v at once.
Altogether: N6 log N.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 174: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/174.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
The Algorithm
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2
Φ(v , ω) = F−1 [ϕ(ξ)d(ξ · ω)] (v , ω)]
Ψ(v , ω) = F−1[ϕ(ξ)d(ξ · ω1)d(ξ · ω2)](v , ω1, ω2)]
1 Fourier transform f → ϕ2 For each node ω1 of a quadrature rule on S2:
1 For each node ω2 of a quadrature rule on ω⊥1 :
1 Compute FΦ(ξ) := ϕ(ξ)d(ξ · ω).2 Compute FΨ(ξ) := ϕ(ξ)d(ξ · ω1)d(ξ · ω2).3 Compute the inverse Fourier transform of FΦ and FΨ.
3 Add up contributions, giving Q(f , f ) for all v at once.
Altogether: N6 log N.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 175: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/175.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
The Algorithm
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2
Φ(v , ω) = F−1 [ϕ(ξ)d(ξ · ω)] (v , ω)]
Ψ(v , ω) = F−1[ϕ(ξ)d(ξ · ω1)d(ξ · ω2)](v , ω1, ω2)]
1 Fourier transform f → ϕ2 For each node ω1 of a quadrature rule on S2:
1 For each node ω2 of a quadrature rule on ω⊥1 :
1 Compute FΦ(ξ) := ϕ(ξ)d(ξ · ω).2 Compute FΨ(ξ) := ϕ(ξ)d(ξ · ω1)d(ξ · ω2).3 Compute the inverse Fourier transform of FΦ and FΨ.
3 Add up contributions, giving Q(f , f ) for all v at once.
Altogether: N6 log N.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 176: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/176.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
The Algorithm
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2
Φ(v , ω) = F−1 [ϕ(ξ)d(ξ · ω)] (v , ω)]
Ψ(v , ω) = F−1[ϕ(ξ)d(ξ · ω1)d(ξ · ω2)](v , ω1, ω2)]
1 Fourier transform f → ϕ2 For each node ω1 of a quadrature rule on S2:
1 For each node ω2 of a quadrature rule on ω⊥1 :1 Compute FΦ(ξ) := ϕ(ξ)d(ξ · ω).
2 Compute FΨ(ξ) := ϕ(ξ)d(ξ · ω1)d(ξ · ω2).3 Compute the inverse Fourier transform of FΦ and FΨ.
3 Add up contributions, giving Q(f , f ) for all v at once.
Altogether: N6 log N.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 177: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/177.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
The Algorithm
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2
Φ(v , ω) = F−1 [ϕ(ξ)d(ξ · ω)] (v , ω)]
Ψ(v , ω) = F−1[ϕ(ξ)d(ξ · ω1)d(ξ · ω2)](v , ω1, ω2)]
1 Fourier transform f → ϕ2 For each node ω1 of a quadrature rule on S2:
1 For each node ω2 of a quadrature rule on ω⊥1 :1 Compute FΦ(ξ) := ϕ(ξ)d(ξ · ω).2 Compute FΨ(ξ) := ϕ(ξ)d(ξ · ω1)d(ξ · ω2).
3 Compute the inverse Fourier transform of FΦ and FΨ.
3 Add up contributions, giving Q(f , f ) for all v at once.
Altogether: N6 log N.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 178: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/178.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
The Algorithm
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2
Φ(v , ω) = F−1 [ϕ(ξ)d(ξ · ω)] (v , ω)]
Ψ(v , ω) = F−1[ϕ(ξ)d(ξ · ω1)d(ξ · ω2)](v , ω1, ω2)]
1 Fourier transform f → ϕ2 For each node ω1 of a quadrature rule on S2:
1 For each node ω2 of a quadrature rule on ω⊥1 :1 Compute FΦ(ξ) := ϕ(ξ)d(ξ · ω).2 Compute FΨ(ξ) := ϕ(ξ)d(ξ · ω1)d(ξ · ω2).3 Compute the inverse Fourier transform of FΦ and FΨ.
3 Add up contributions, giving Q(f , f ) for all v at once.
Altogether: N6 log N.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 179: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/179.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
The Algorithm
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2
Φ(v , ω) = F−1 [ϕ(ξ)d(ξ · ω)] (v , ω)]
Ψ(v , ω) = F−1[ϕ(ξ)d(ξ · ω1)d(ξ · ω2)](v , ω1, ω2)]
1 Fourier transform f → ϕ2 For each node ω1 of a quadrature rule on S2:
1 For each node ω2 of a quadrature rule on ω⊥1 :1 Compute FΦ(ξ) := ϕ(ξ)d(ξ · ω).2 Compute FΨ(ξ) := ϕ(ξ)d(ξ · ω1)d(ξ · ω2).3 Compute the inverse Fourier transform of FΦ and FΨ.
3 Add up contributions, giving Q(f , f ) for all v at once.
Altogether: N6 log N.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 180: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/180.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Bobylev-Rjasanow’s Integral Transform Method
The Algorithm
Q(f , f ) =
∫S2
∫S2
δ(ω1 · ω2) [Φ(ω1)Φ(ω2)− f Ψ(ω1, ω2)] dω1dω2
Φ(v , ω) = F−1 [ϕ(ξ)d(ξ · ω)] (v , ω)]
Ψ(v , ω) = F−1[ϕ(ξ)d(ξ · ω1)d(ξ · ω2)](v , ω1, ω2)]
1 Fourier transform f → ϕ2 For each node ω1 of a quadrature rule on S2:
1 For each node ω2 of a quadrature rule on ω⊥1 :1 Compute FΦ(ξ) := ϕ(ξ)d(ξ · ω).2 Compute FΨ(ξ) := ϕ(ξ)d(ξ · ω1)d(ξ · ω2).3 Compute the inverse Fourier transform of FΦ and FΨ.
3 Add up contributions, giving Q(f , f ) for all v at once.
Altogether: N6 log N.A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 181: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/181.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Pareschi-Russo’s Spectral Method
Idea:
Do Fourier-Galerkin on ∂t f = Q(f , f ).Demand that the residual of ∂t fN = Q(fN , fN) is orthogonal to alltrigonometric polynomials, where fN is a Fourier expansion.
skip the details
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Pareschi-Russo’s Spectral Method
Expand
fN(v) :=∑
k∈K3N
fke i(k·v)
with Fourier coefficients
fk =1
(2π)3
∫[−π,π]3
f (v)e−i(k·v)dv .
Assume suppv (f ) ⊂ B(0,R0). Take KN := {−N, . . . ,N}.As before, work on [−L, L]3. WLOG π = L.
Avoid aliasing: π = L >3 +√
2
2R0.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 183: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/183.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Pareschi-Russo’s Spectral Method
Expand
fN(v) :=∑
k∈K3N
fke i(k·v)
with Fourier coefficients
fk =1
(2π)3
∫[−π,π]3
f (v)e−i(k·v)dv .
Assume suppv (f ) ⊂ B(0,R0). Take KN := {−N, . . . ,N}.As before, work on [−L, L]3. WLOG π = L.
Avoid aliasing: π = L >3 +√
2
2R0.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 184: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/184.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Pareschi-Russo’s Spectral Method
Expand
fN(v) :=∑
k∈K3N
fke i(k·v)
with Fourier coefficients
fk =1
(2π)3
∫[−π,π]3
f (v)e−i(k·v)dv .
Assume suppv (f ) ⊂ B(0,R0).
Take KN := {−N, . . . ,N}.As before, work on [−L, L]3. WLOG π = L.
Avoid aliasing: π = L >3 +√
2
2R0.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 185: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/185.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Pareschi-Russo’s Spectral Method
Expand
fN(v) :=∑
k∈K3N
fke i(k·v)
with Fourier coefficients
fk =1
(2π)3
∫[−π,π]3
f (v)e−i(k·v)dv .
Assume suppv (f ) ⊂ B(0,R0). Take KN := {−N, . . . ,N}.
As before, work on [−L, L]3. WLOG π = L.
Avoid aliasing: π = L >3 +√
2
2R0.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 186: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/186.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Pareschi-Russo’s Spectral Method
Expand
fN(v) :=∑
k∈K3N
fke i(k·v)
with Fourier coefficients
fk =1
(2π)3
∫[−π,π]3
f (v)e−i(k·v)dv .
Assume suppv (f ) ⊂ B(0,R0). Take KN := {−N, . . . ,N}.As before, work on [−L, L]3.
WLOG π = L.
Avoid aliasing: π = L >3 +√
2
2R0.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 187: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/187.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Pareschi-Russo’s Spectral Method
Expand
fN(v) :=∑
k∈K3N
fke i(k·v)
with Fourier coefficients
fk =1
(2π)3
∫[−π,π]3
f (v)e−i(k·v)dv .
Assume suppv (f ) ⊂ B(0,R0). Take KN := {−N, . . . ,N}.As before, work on [−L, L]3. WLOG π = L.
Avoid aliasing: π = L >3 +√
2
2R0.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 188: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/188.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Pareschi-Russo’s Spectral Method
Expand
fN(v) :=∑
k∈K3N
fke i(k·v)
with Fourier coefficients
fk =1
(2π)3
∫[−π,π]3
f (v)e−i(k·v)dv .
Assume suppv (f ) ⊂ B(0,R0). Take KN := {−N, . . . ,N}.As before, work on [−L, L]3. WLOG π = L.
Avoid aliasing: π = L >3 +√
2
2R0.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 189: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/189.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Pareschi-Russo’s Spectral Method
Residual of ∂t fN = Q(fN , fN) is L2-orthogonal to all trigonometricpolynomials:
∫[−π,π]3
(∂fN∂t− Q(fN , fN)
)e−ik·vdv
!= 0 k ∈ K 3
N
→ get the scheme
∂ fk∂t
=∑
m∈K3N
fk−m fmB(k −m,m)−∑
m∈K3N
fk−m fmB(m,m)
with the kernel modes
B(l ,m) =
∫B(0,2R0)
∫S2
B(|g |, cos θ)e−ig · (l+m)2−i |g |ω· (m−l)
2 dωdg
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 190: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/190.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Pareschi-Russo’s Spectral Method
Residual of ∂t fN = Q(fN , fN) is L2-orthogonal to all trigonometricpolynomials:∫
[−π,π]3
(∂fN∂t− Q(fN , fN)
)e−ik·vdv
!= 0 k ∈ K 3
N
→ get the scheme
∂ fk∂t
=∑
m∈K3N
fk−m fmB(k −m,m)−∑
m∈K3N
fk−m fmB(m,m)
with the kernel modes
B(l ,m) =
∫B(0,2R0)
∫S2
B(|g |, cos θ)e−ig · (l+m)2−i |g |ω· (m−l)
2 dωdg
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 191: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/191.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Pareschi-Russo’s Spectral Method
Residual of ∂t fN = Q(fN , fN) is L2-orthogonal to all trigonometricpolynomials:∫
[−π,π]3
(∂fN∂t− Q(fN , fN)
)e−ik·vdv
!= 0 k ∈ K 3
N
→ get the scheme
∂ fk∂t
=∑
m∈K3N
fk−m fmB(k −m,m)−∑
m∈K3N
fk−m fmB(m,m)
with the kernel modes
B(l ,m) =
∫B(0,2R0)
∫S2
B(|g |, cos θ)e−ig · (l+m)2−i |g |ω· (m−l)
2 dωdg
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 192: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/192.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Pareschi-Russo’s Spectral Method
Residual of ∂t fN = Q(fN , fN) is L2-orthogonal to all trigonometricpolynomials:∫
[−π,π]3
(∂fN∂t− Q(fN , fN)
)e−ik·vdv
!= 0 k ∈ K 3
N
→ get the scheme
∂ fk∂t
=∑
m∈K3N
fk−m fmB(k −m,m)−∑
m∈K3N
fk−m fmB(m,m)
with the kernel modes
B(l ,m) =
∫B(0,2R0)
∫S2
B(|g |, cos θ)e−ig · (l+m)2−i |g |ω· (m−l)
2 dωdg
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 193: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/193.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Performance of the Pareschi-Russo Spectral Scheme
∂ fk∂t
=∑
m∈K3N
fk−m fmB(k −m,m)−∑
m∈K3N
fk−m fmB(m,m)
Evaluation of the scheme takes O(N6) operations.
Because we can pre-evaluate B.Recall: naive method would take O(N6Na).
Big plus: Rolled evaluation of problematic sphere integral intopre-computation.
Loss term is sum of convolutions.→ Can evaluate in O(N3 log N).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 194: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/194.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Performance of the Pareschi-Russo Spectral Scheme
∂ fk∂t
=∑
m∈K3N
fk−m fmB(k −m,m)−∑
m∈K3N
fk−m fmB(m,m)
Evaluation of the scheme takes O(N6) operations.
Because we can pre-evaluate B.
Recall: naive method would take O(N6Na).
Big plus: Rolled evaluation of problematic sphere integral intopre-computation.
Loss term is sum of convolutions.→ Can evaluate in O(N3 log N).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 195: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/195.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Performance of the Pareschi-Russo Spectral Scheme
∂ fk∂t
=∑
m∈K3N
fk−m fmB(k −m,m)−∑
m∈K3N
fk−m fmB(m,m)
Evaluation of the scheme takes O(N6) operations.
Because we can pre-evaluate B.Recall: naive method would take O(N6Na).
Big plus: Rolled evaluation of problematic sphere integral intopre-computation.
Loss term is sum of convolutions.→ Can evaluate in O(N3 log N).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 196: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/196.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Performance of the Pareschi-Russo Spectral Scheme
∂ fk∂t
=∑
m∈K3N
fk−m fmB(k −m,m)−∑
m∈K3N
fk−m fmB(m,m)
Evaluation of the scheme takes O(N6) operations.
Because we can pre-evaluate B.Recall: naive method would take O(N6Na).
Big plus: Rolled evaluation of problematic sphere integral intopre-computation.
Loss term is sum of convolutions.→ Can evaluate in O(N3 log N).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Performance of the Pareschi-Russo Spectral Scheme
∂ fk∂t
=∑
m∈K3N
fk−m fmB(k −m,m)−∑
m∈K3N
fk−m fmB(m,m)
Evaluation of the scheme takes O(N6) operations.
Because we can pre-evaluate B.Recall: naive method would take O(N6Na).
Big plus: Rolled evaluation of problematic sphere integral intopre-computation.
Loss term is sum of convolutions.→ Can evaluate in O(N3 log N).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Kernel modes of the Pareschi-Russo Spectral Scheme
B(l ,m) =
∫B(0,2R0)
∫S2
B(|g |, cos θ)e−ig · (l+m)2−i |g |ω· (m−l)
2 dωdg
The pre-evaluated B takes up only limited storage.
In the general case, B depends on |l −m|, |l + m|,(l −m) · (l + m) .For VHS, B depends only on |l −m|, |l + m|.→ Two-dimensional array suffices.
A far more explicit expression for B is available.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Kernel modes of the Pareschi-Russo Spectral Scheme
B(l ,m) =
∫B(0,2R0)
∫S2
B(|g |, cos θ)e−ig · (l+m)2−i |g |ω· (m−l)
2 dωdg
The pre-evaluated B takes up only limited storage.
In the general case, B depends on |l −m|, |l + m|,(l −m) · (l + m) .
For VHS, B depends only on |l −m|, |l + m|.→ Two-dimensional array suffices.
A far more explicit expression for B is available.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Kernel modes of the Pareschi-Russo Spectral Scheme
B(l ,m) =
∫B(0,2R0)
∫S2
B(|g |, cos θ)e−ig · (l+m)2−i |g |ω· (m−l)
2 dωdg
The pre-evaluated B takes up only limited storage.
In the general case, B depends on |l −m|, |l + m|,(l −m) · (l + m) .For VHS, B depends only on |l −m|, |l + m|.→ Two-dimensional array suffices.
A far more explicit expression for B is available.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 201: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/201.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Kernel modes of the Pareschi-Russo Spectral Scheme
B(l ,m) =
∫B(0,2R0)
∫S2
B(|g |, cos θ)e−ig · (l+m)2−i |g |ω· (m−l)
2 dωdg
The pre-evaluated B takes up only limited storage.
In the general case, B depends on |l −m|, |l + m|,(l −m) · (l + m) .For VHS, B depends only on |l −m|, |l + m|.→ Two-dimensional array suffices.
A far more explicit expression for B is available.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 202: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/202.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Analysis of the Pareschi-Russo Spectral Scheme
Theorem (Spectral Accuracy)
Let f ∈ L2([−π, π]3). Then
∥∥Qtr(f , f )− PNQtr(f , f )∥∥
26 C
(‖f − fN‖2 +
‖Qtr(fN , fN)‖Hrper
N r
)
for all r > 0.
Can show stability after including some filtering in the scheme.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Pareschi-Russo’s Spectral Method
Analysis of the Pareschi-Russo Spectral Scheme
Theorem (Spectral Accuracy)
Let f ∈ L2([−π, π]3). Then
∥∥Qtr(f , f )− PNQtr(f , f )∥∥
26 C
(‖f − fN‖2 +
‖Qtr(fN , fN)‖Hrper
N r
)
for all r > 0.
Can show stability after including some filtering in the scheme.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Mouhot-Pareschi’s Spectral Method
Idea:
Combine the symmetric representation from Bobylev-Rjasanowwith the Fourier-Galerkin approach from Pareschi-Russo.
skip the details
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Symmetric Representation
Recall
Lemma (Symmetric Representation)
The collision operator can be represented as
Q(f , f ) =
∫Rd
∫Rd
B(y , z)δ(z · y)[f (v + z)f (v + y)
− f (v)f (v + y + z)]dydz ,
with
B(y , z) :=2d−1
|y + z |d−2B
(|y + z |, y · (y + z)
|y ||y + z |).
(slight generalization for multi-d)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Symmetric Representation
Recall
Lemma (Symmetric Representation)
The collision operator can be represented as
Q(f , f ) =
∫Rd
∫Rd
B(y , z)δ(z · y)[f (v + z)f (v + y)
− f (v)f (v + y + z)]dydz ,
with
B(y , z) :=2d−1
|y + z |d−2B
(|y + z |, y · (y + z)
|y ||y + z |).
(slight generalization for multi-d)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Truncating the Symmetric Representation
As before, assume suppv (f ) ⊂ B(0,R0).
Work on [−L, L]3.WLOG π = L.
Avoid aliasing: π = L >3 +√
2
2R0.
Then y and z may be limited to B(0,R), with R :=√
2R0:
QR(f , f ) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[f (v + z)f (v + y)
− f (v)f (v + y + z)]dydz
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Truncating the Symmetric Representation
As before, assume suppv (f ) ⊂ B(0,R0). Work on [−L, L]3.
WLOG π = L.
Avoid aliasing: π = L >3 +√
2
2R0.
Then y and z may be limited to B(0,R), with R :=√
2R0:
QR(f , f ) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[f (v + z)f (v + y)
− f (v)f (v + y + z)]dydz
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Truncating the Symmetric Representation
As before, assume suppv (f ) ⊂ B(0,R0). Work on [−L, L]3.WLOG π = L.
Avoid aliasing: π = L >3 +√
2
2R0.
Then y and z may be limited to B(0,R), with R :=√
2R0:
QR(f , f ) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[f (v + z)f (v + y)
− f (v)f (v + y + z)]dydz
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Truncating the Symmetric Representation
As before, assume suppv (f ) ⊂ B(0,R0). Work on [−L, L]3.WLOG π = L.
Avoid aliasing: π = L >3 +√
2
2R0.
Then y and z may be limited to B(0,R), with R :=√
2R0:
QR(f , f ) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[f (v + z)f (v + y)
− f (v)f (v + y + z)]dydz
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Truncating the Symmetric Representation
As before, assume suppv (f ) ⊂ B(0,R0). Work on [−L, L]3.WLOG π = L.
Avoid aliasing: π = L >3 +√
2
2R0.
Then y and z may be limited to B(0,R), with R :=√
2R0:
QR(f , f ) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[f (v + z)f (v + y)
− f (v)f (v + y + z)]dydz
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Truncating the Symmetric Representation
As before, assume suppv (f ) ⊂ B(0,R0). Work on [−L, L]3.WLOG π = L.
Avoid aliasing: π = L >3 +√
2
2R0.
Then y and z may be limited to B(0,R), with R :=√
2R0:
QR(f , f ) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[f (v + z)f (v + y)
− f (v)f (v + y + z)]dydz
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0!
=
∫[−π,π]d
(∂fN∂t− [Q+(fN , fN)− L(fN)fN
])e−ik·vdv
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0!
=
∫[−π,π]d
(∂fN∂t− [Q+(fN , fN)− L(fN)fN
])e−ik·vdv
=
∫[−π,π]d
∑k ′∈Kd
N
∂ fk ′
∂te i(k ′·v) − QR(fN , fN)
e−ik·vdv
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0 =
∫[−π,π]d
∑k ′∈Kd
N
∂ fk ′
∂te i(k ′·v) − QR(fN , fN)
e−ik·vdv
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0 =
∫[−π,π]d
∑k ′∈Kd
N
∂ fk ′
∂te i(k ′·v) − QR(fN , fN)
e−ik·vdv
=∂ fk∂t−∫
[−π,π]d
[QR(fN , fN)
]e−ik·vdv
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0 =∂ fk∂t−∫
[−π,π]d
[QR(fN , fN)
]e−ik·vdv
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0 =∂ fk∂t−∫
[−π,π]d
[QR(fN , fN)
]e−ik·vdv
=∂ fk∂t−∫
[−π,π]d
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[fN(v + z)fN(v + y)
− fN(v)fN(v + y + z)]e−ik·vdydzdv
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0 =∂ fk∂t−∫
[−π,π]d
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[fN(v + z)fN(v + y)
− fN(v)fN(v + y + z)]e−ik·vdydzdv
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0 =∂ fk∂t−∫
[−π,π]d
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[fN(v + z)fN(v + y)
− fN(v)fN(v + y + z)]e−ik·vdydzdv
=∂ fk∂t−∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)
∫[−π,π]d∑
l ,m∈K3N
fl fme i [l ·(v+z)+m·(v+y)−k·v ]
−∑
l ,m∈K3N
fl fme i [l ·v+m·(v+y+z)−k·v ]dvdydz
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0 =∂ fk∂t−∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)
∫[−π,π]d∑
l ,m∈K3N
fl fme i [l ·(v+z)+m·(v+y)−k·v ]
−∑
l ,m∈K3N
fl fme i [l ·v+m·(v+y+z)−k·v ]dvdydz
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0 =∂ fk∂t−∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)
∫[−π,π]d∑
l ,m∈K3N
fl fme i [l ·(v+z)+m·(v+y)−k·v ]
−∑
l ,m∈K3N
fl fme i [l ·v+m·(v+y+z)−k·v ]dvdydz
=∂ fk∂t−∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)∑l ,m∈K3
N
fl fmδl+m,k
[e i [l ·z+m·y ] − e im·(y+z)
]dydz
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0 =∂ fk∂t−∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)∑l ,m∈K3
N
fl fmδl+m,k
[e i [l ·z+m·y ] − e im·(y+z)
]dydz
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0 =∂ fk∂t−∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)∑l ,m∈K3
N
fl fmδl+m,k
[e i [l ·z+m·y ] − e im·(y+z)
]dydz
=∂ fk∂t−
∑l ,m∈K3
N ,l+m=k
fl fm
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)
[e i [l ·z+m·y ] − e im·(y+z)
]dydz
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0 =∂ fk∂t−
∑l ,m∈K3
N ,l+m=k
fl fm
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)
[e i [l ·z+m·y ] − e im·(y+z)
]dydz
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0 =∂ fk∂t−
∑l ,m∈K3
N ,l+m=k
fl fm
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)
[e i [l ·z+m·y ] − e im·(y+z)
]dydz
=∂ fk∂t−
∑l ,m∈K3
N ,l+m=k
fl fmβ(l ,m)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0 =∂ fk∂t−
∑l ,m∈K3
N ,l+m=k
fl fmβ(l ,m)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
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Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Derivation
skip the details
0 =∂ fk∂t−
∑l ,m∈K3
N ,l+m=k
fl fmβ(l ,m)
=∂ fk∂t−∑
m∈K3N
fk−m fmβ(k −m,m)
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 229: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/229.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Summary
We obtained
∂ fk∂t
=∑
m∈K3N
fk−m fmβ(k −m,m)
with
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z ·y)[e i [l ·z+m·y ] − e im·(y+z)
]dydz .
Simplify by setting
β(l ,m) = β(l ,m)− β(m,m)
with
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z ·y)[e i [l ·y+m·z]
]dydz = β(m, l).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 230: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/230.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Summary
We obtained
∂ fk∂t
=∑
m∈K3N
fk−m fmβ(k −m,m)
with
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z ·y)[e i [l ·z+m·y ] − e im·(y+z)
]dydz .
Simplify by setting
β(l ,m) = β(l ,m)− β(m,m)
with
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z ·y)[e i [l ·y+m·z]
]dydz = β(m, l).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 231: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/231.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Summary
We obtained
∂ fk∂t
=∑
m∈K3N
fk−m fmβ(k −m,m)
with
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z ·y)[e i [l ·z+m·y ] − e im·(y+z)
]dydz .
Simplify by setting
β(l ,m) = β(l ,m)− β(m,m)
with
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z ·y)[e i [l ·y+m·z]
]dydz = β(m, l).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 232: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/232.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fourier-Galerkin Summary
We obtained
∂ fk∂t
=∑
m∈K3N
fk−m fmβ(k −m,m)
with
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z ·y)[e i [l ·z+m·y ] − e im·(y+z)
]dydz .
Simplify by setting
β(l ,m) = β(l ,m)− β(m,m)
with
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z ·y)[e i [l ·y+m·z]
]dydz = β(m, l).
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 233: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/233.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Remarks on the Scheme
∂ fk∂t
=∑
m∈K3N
fk−m fm[β(k −m,m)− β(m,m)]
Pareschi-Russo scheme looked exactly the same.
(but with different β)→ How are we going to save time here?Recall: Pareschi-Russo loss term could be evaluated inO(N3 log N) because of special structure.→ Try for the same here.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 234: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/234.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Remarks on the Scheme
∂ fk∂t
=∑
m∈K3N
fk−m fm[β(k −m,m)− β(m,m)]
Pareschi-Russo scheme looked exactly the same.(but with different β)
→ How are we going to save time here?Recall: Pareschi-Russo loss term could be evaluated inO(N3 log N) because of special structure.→ Try for the same here.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 235: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/235.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Remarks on the Scheme
∂ fk∂t
=∑
m∈K3N
fk−m fm[β(k −m,m)− β(m,m)]
Pareschi-Russo scheme looked exactly the same.(but with different β)→ How are we going to save time here?
Recall: Pareschi-Russo loss term could be evaluated inO(N3 log N) because of special structure.→ Try for the same here.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 236: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/236.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Remarks on the Scheme
∂ fk∂t
=∑
m∈K3N
fk−m fm[β(k −m,m)− β(m,m)]
Pareschi-Russo scheme looked exactly the same.(but with different β)→ How are we going to save time here?Recall: Pareschi-Russo loss term could be evaluated inO(N3 log N) because of special structure.
→ Try for the same here.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 237: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/237.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Remarks on the Scheme
∂ fk∂t
=∑
m∈K3N
fk−m fm[β(k −m,m)− β(m,m)]
Pareschi-Russo scheme looked exactly the same.(but with different β)→ How are we going to save time here?Recall: Pareschi-Russo loss term could be evaluated inO(N3 log N) because of special structure.→ Try for the same here.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 238: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/238.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Special Structure of the Kernel Modes
For Hard Spheres in 3D, we obtain the special structure
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[e i [l ·y+m·z]
]dydz
≈ β(l ,m) :=π2
M2
M∑p,q=0
αp,q(l)α′p,q(m)
How? skip (roughly)
Introduce spherical coordinates.Assume decoupling B(y , z) = a(|y |)b(|z |) for y ⊥ z . (Holdsfor Hard Spheres in 3D, Maxwell molecules in 2D, . . . )Periodicity reduces spherical integration to hemisphere S2
+.Parameterize S2
+ by (θ, ϕ) ∈ [0, π)2.Discretize
θ := pπ/M, ϕ := qπ/M.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 239: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/239.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Special Structure of the Kernel Modes
For Hard Spheres in 3D, we obtain the special structure
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[e i [l ·y+m·z]
]dydz
≈ β(l ,m) :=π2
M2
M∑p,q=0
αp,q(l)α′p,q(m)
How? skip (roughly)
Introduce spherical coordinates.Assume decoupling B(y , z) = a(|y |)b(|z |) for y ⊥ z . (Holdsfor Hard Spheres in 3D, Maxwell molecules in 2D, . . . )Periodicity reduces spherical integration to hemisphere S2
+.Parameterize S2
+ by (θ, ϕ) ∈ [0, π)2.Discretize
θ := pπ/M, ϕ := qπ/M.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 240: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/240.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Special Structure of the Kernel Modes
For Hard Spheres in 3D, we obtain the special structure
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[e i [l ·y+m·z]
]dydz
≈ β(l ,m) :=π2
M2
M∑p,q=0
αp,q(l)α′p,q(m)
How? skip (roughly)
Introduce spherical coordinates.Assume decoupling B(y , z) = a(|y |)b(|z |) for y ⊥ z . (Holdsfor Hard Spheres in 3D, Maxwell molecules in 2D, . . . )Periodicity reduces spherical integration to hemisphere S2
+.Parameterize S2
+ by (θ, ϕ) ∈ [0, π)2.Discretize
θ := pπ/M, ϕ := qπ/M.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 241: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/241.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Special Structure of the Kernel Modes
For Hard Spheres in 3D, we obtain the special structure
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[e i [l ·y+m·z]
]dydz
≈ β(l ,m) :=π2
M2
M∑p,q=0
αp,q(l)α′p,q(m)
How? skip
(roughly)
Introduce spherical coordinates.Assume decoupling B(y , z) = a(|y |)b(|z |) for y ⊥ z . (Holdsfor Hard Spheres in 3D, Maxwell molecules in 2D, . . . )Periodicity reduces spherical integration to hemisphere S2
+.Parameterize S2
+ by (θ, ϕ) ∈ [0, π)2.Discretize
θ := pπ/M, ϕ := qπ/M.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 242: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/242.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Special Structure of the Kernel Modes
For Hard Spheres in 3D, we obtain the special structure
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[e i [l ·y+m·z]
]dydz
≈ β(l ,m) :=π2
M2
M∑p,q=0
αp,q(l)α′p,q(m)
How? skip (roughly)
Introduce spherical coordinates.
Assume decoupling B(y , z) = a(|y |)b(|z |) for y ⊥ z . (Holdsfor Hard Spheres in 3D, Maxwell molecules in 2D, . . . )Periodicity reduces spherical integration to hemisphere S2
+.Parameterize S2
+ by (θ, ϕ) ∈ [0, π)2.Discretize
θ := pπ/M, ϕ := qπ/M.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 243: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/243.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Special Structure of the Kernel Modes
For Hard Spheres in 3D, we obtain the special structure
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[e i [l ·y+m·z]
]dydz
≈ β(l ,m) :=π2
M2
M∑p,q=0
αp,q(l)α′p,q(m)
How? skip (roughly)
Introduce spherical coordinates.Assume decoupling B(y , z) = a(|y |)b(|z |) for y ⊥ z . (Holdsfor Hard Spheres in 3D, Maxwell molecules in 2D, . . . )
Periodicity reduces spherical integration to hemisphere S2+.
Parameterize S2+ by (θ, ϕ) ∈ [0, π)2.
Discretizeθ := pπ/M, ϕ := qπ/M.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 244: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/244.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Special Structure of the Kernel Modes
For Hard Spheres in 3D, we obtain the special structure
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[e i [l ·y+m·z]
]dydz
≈ β(l ,m) :=π2
M2
M∑p,q=0
αp,q(l)α′p,q(m)
How? skip (roughly)
Introduce spherical coordinates.Assume decoupling B(y , z) = a(|y |)b(|z |) for y ⊥ z . (Holdsfor Hard Spheres in 3D, Maxwell molecules in 2D, . . . )Periodicity reduces spherical integration to hemisphere S2
+.
Parameterize S2+ by (θ, ϕ) ∈ [0, π)2.
Discretizeθ := pπ/M, ϕ := qπ/M.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 245: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/245.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Special Structure of the Kernel Modes
For Hard Spheres in 3D, we obtain the special structure
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[e i [l ·y+m·z]
]dydz
≈ β(l ,m) :=π2
M2
M∑p,q=0
αp,q(l)α′p,q(m)
How? skip (roughly)
Introduce spherical coordinates.Assume decoupling B(y , z) = a(|y |)b(|z |) for y ⊥ z . (Holdsfor Hard Spheres in 3D, Maxwell molecules in 2D, . . . )Periodicity reduces spherical integration to hemisphere S2
+.Parameterize S2
+ by (θ, ϕ) ∈ [0, π)2.
Discretizeθ := pπ/M, ϕ := qπ/M.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 246: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/246.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Special Structure of the Kernel Modes
For Hard Spheres in 3D, we obtain the special structure
β(l ,m) :=
∫B(0,R)
∫B(0,R)
B(y , z)δ(z · y)[e i [l ·y+m·z]
]dydz
≈ β(l ,m) :=π2
M2
M∑p,q=0
αp,q(l)α′p,q(m)
How? skip (roughly)
Introduce spherical coordinates.Assume decoupling B(y , z) = a(|y |)b(|z |) for y ⊥ z . (Holdsfor Hard Spheres in 3D, Maxwell molecules in 2D, . . . )Periodicity reduces spherical integration to hemisphere S2
+.Parameterize S2
+ by (θ, ϕ) ∈ [0, π)2.Discretize
θ := pπ/M, ϕ := qπ/M.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 247: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/247.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fast Evaluation of the Scheme
∂ fk∂t
=π2
M2
∑l ,m∈K3
N
fk−m fmβ(k −m,m) − loss
=π2
M2
∑m∈K3
N
fk−m fm
M∑p,q=0
αp,q(k −m)α′p,q(m) − loss
=π2
M2
M∑p,q=0
∑m∈K3
N
fk−mαp,q(k −m)fmα′p,q(m) − loss.
→ Sum of M2 convolutions. Use FFT: M2N3 log N operations.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 248: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/248.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fast Evaluation of the Scheme
∂ fk∂t
=π2
M2
∑l ,m∈K3
N
fk−m fmβ(k −m,m) − loss
=π2
M2
∑m∈K3
N
fk−m fm
M∑p,q=0
αp,q(k −m)α′p,q(m) − loss
=π2
M2
M∑p,q=0
∑m∈K3
N
fk−mαp,q(k −m)fmα′p,q(m) − loss.
→ Sum of M2 convolutions. Use FFT: M2N3 log N operations.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 249: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/249.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fast Evaluation of the Scheme
∂ fk∂t
=π2
M2
∑l ,m∈K3
N
fk−m fmβ(k −m,m) − loss
=π2
M2
∑m∈K3
N
fk−m fm
M∑p,q=0
αp,q(k −m)α′p,q(m) − loss
=π2
M2
M∑p,q=0
∑m∈K3
N
fk−mαp,q(k −m)fmα′p,q(m) − loss.
→ Sum of M2 convolutions. Use FFT: M2N3 log N operations.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 250: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/250.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fast Evaluation of the Scheme
∂ fk∂t
=π2
M2
∑l ,m∈K3
N
fk−m fmβ(k −m,m) − loss
=π2
M2
∑m∈K3
N
fk−m fm
M∑p,q=0
αp,q(k −m)α′p,q(m) − loss
=π2
M2
M∑p,q=0
∑m∈K3
N
fk−mαp,q(k −m)fmα′p,q(m) − loss.
→ Sum of M2 convolutions.
Use FFT: M2N3 log N operations.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 251: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/251.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Fast Evaluation of the Scheme
∂ fk∂t
=π2
M2
∑l ,m∈K3
N
fk−m fmβ(k −m,m) − loss
=π2
M2
∑m∈K3
N
fk−m fm
M∑p,q=0
αp,q(k −m)α′p,q(m) − loss
=π2
M2
M∑p,q=0
∑m∈K3
N
fk−mαp,q(k −m)fmα′p,q(m) − loss.
→ Sum of M2 convolutions. Use FFT: M2N3 log N operations.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 252: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/252.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Final Remarks on Mouhot-Pareschi
∂ fk∂t
=π2
M2
M∑p,q=0
∑m∈K3
N
fk−mαp,q(k −m)fmα′p,q(m) − loss.
Fastest method around.
Spectrally accurate. (Similar result as for Pareschi-Russo)
Only “sophisticated” numerical technique is FFT. Probablythe easiest to implement of all three methods discussed.
Some care needs to be taken to retain proper conservation.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 253: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/253.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Final Remarks on Mouhot-Pareschi
∂ fk∂t
=π2
M2
M∑p,q=0
∑m∈K3
N
fk−mαp,q(k −m)fmα′p,q(m) − loss.
Fastest method around.
Spectrally accurate. (Similar result as for Pareschi-Russo)
Only “sophisticated” numerical technique is FFT. Probablythe easiest to implement of all three methods discussed.
Some care needs to be taken to retain proper conservation.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 254: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/254.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Final Remarks on Mouhot-Pareschi
∂ fk∂t
=π2
M2
M∑p,q=0
∑m∈K3
N
fk−mαp,q(k −m)fmα′p,q(m) − loss.
Fastest method around.
Spectrally accurate. (Similar result as for Pareschi-Russo)
Only “sophisticated” numerical technique is FFT. Probablythe easiest to implement of all three methods discussed.
Some care needs to be taken to retain proper conservation.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 255: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/255.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Mouhot-Pareschi’s Sub-N6 Method
Final Remarks on Mouhot-Pareschi
∂ fk∂t
=π2
M2
M∑p,q=0
∑m∈K3
N
fk−mαp,q(k −m)fmα′p,q(m) − loss.
Fastest method around.
Spectrally accurate. (Similar result as for Pareschi-Russo)
Only “sophisticated” numerical technique is FFT. Probablythe easiest to implement of all three methods discussed.
Some care needs to be taken to retain proper conservation.
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 256: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/256.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Conclusion
Conclusion
M H V O Order Ops Ms Mm En
DSMC � � � � N−1/2 N3 � � �IbrR � � � − N−2 N6 � ◦ ◦
gen. − − − � N−2 N6 log N � ◦ ◦BobR − 3 − − N−2 N6 log N � ◦ ◦ParRu � � � − N−∞ N6 � ◦ ◦MouP 2 3 − − N−∞ N5 log N � ◦ ◦
Additional topic: Wild expansion timestepping for the collisionoperator.
maintains positivity
see notes
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 257: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/257.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Conclusion
Conclusion
M H V O Order Ops Ms Mm En
DSMC � � � � N−1/2 N3 � � �IbrR � � � − N−2 N6 � ◦ ◦
gen. − − − � N−2 N6 log N � ◦ ◦BobR − 3 − − N−2 N6 log N � ◦ ◦ParRu � � � − N−∞ N6 � ◦ ◦MouP 2 3 − − N−∞ N5 log N � ◦ ◦
Additional topic: Wild expansion timestepping for the collisionoperator.
maintains positivity
see notes
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 258: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/258.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Conclusion
Conclusion
M H V O Order Ops Ms Mm En
DSMC � � � � N−1/2 N3 � � �IbrR � � � − N−2 N6 � ◦ ◦
gen. − − − � N−2 N6 log N � ◦ ◦BobR − 3 − − N−2 N6 log N � ◦ ◦ParRu � � � − N−∞ N6 � ◦ ◦MouP 2 3 − − N−∞ N5 log N � ◦ ◦
Additional topic: Wild expansion timestepping for the collisionoperator.
maintains positivity
see notes
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 259: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/259.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Conclusion
Conclusion
M H V O Order Ops Ms Mm En
DSMC � � � � N−1/2 N3 � � �IbrR � � � − N−2 N6 � ◦ ◦
gen. − − − � N−2 N6 log N � ◦ ◦BobR − 3 − − N−2 N6 log N � ◦ ◦ParRu � � � − N−∞ N6 � ◦ ◦MouP 2 3 − − N−∞ N5 log N � ◦ ◦
Additional topic: Wild expansion timestepping for the collisionoperator.
maintains positivity
see notes
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation
![Page 260: Deterministic Methods for the Boltzmann Equation](https://reader036.fdocuments.us/reader036/viewer/2022082914/613c7c46c957d930775e45aa/html5/thumbnails/260.jpg)
Introduction Time-Step Splitting Transport Collision Operator
Conclusion
Questions?
?
A. Narayan and A. Klockner
Deterministic Methods for the Boltzmann Equation