Semi-Lagrangian Formulations for Linear Advection ... · Equations Jingmei Qiu Introduction...

SL for linearadvection withapplications to

kineticEquations

Jingmei Qiu

Introduction

Proposedmethod

SL finitedifference I

SL finitedifference II

Comparison

Simulationresults

Summary

Semi-Lagrangian Formulations for LinearAdvection Equations and Applications to

Kinetic Equations

Jingmei Qiu

Department of Mathematical and Computer ScienceColorado School of Mines

joint work w/ Chi-Wang ShuSupported by NSF and AFOSR.

CSCAMM, University of Maryland College Park

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Outline

• Background: numerical methods for kinetic equations

• Semi-Lagragian finite difference methods for linearadvection equations

• Simulation results

• Ongoing and future work

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

VP system

The Vlasov-Poisson (VP) system,

∂f

∂t+ p · ∇xf + E(t, x) · ∇pf = 0, (1)

E(t, x) = −∇xφ(t, x), −4xφ(t, x) = ρ(t, x). (2)

f (t, x,p): the probability of finding a particle with velocity p atposition x at time t.ρ(t, x) =

∫f (t, x,p)dp - 1: charge density

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Numerical approach:Lagrangian vs. Eulerian vs.

semi-Lagrangian

• Lagrangian: tracking a finite number of macro-particles.– e.g., PIC (Particle In Cell)

dx

dt= v,

dv

dt= E (3)

• Eulerian: fixed numerical mesh– e.g., finite difference WENO, finite volume, finiteelement, spectral method.

• Semi-Lagrangian:–e.g., finite difference, finite volume, finite element,spectral method.

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Strang splitting for solving theVlasov equation

∂f

∂t+ v · ∇xf + E(t, x) · ∇vf = 0,

Nonlinear Vlasov eqation⇒ a sequence of linear equations.

• 1-D in x and 1-D in v :

∂f

∂t+ v

∂f

∂x= 0 , (4)

∂f

∂t+ E (t, x)

∂f

∂v= 0 . (5)

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

SL schemes

1 Solution space: point values, or cell averages, or piecewisepolynomials living on fixed Eulerian grid.

2 Evolution: tracking characteristics backward in time.

3 Project the evolved solution back onto the solution space.

Remark: The only error in time comes from trackingcharacteristics backward in time. The scheme is not subject toCFL time step restriction.

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Various formulationsof SL finite difference schemes

SL finite difference (point values)

• Scheme I: advective schemethe solution is evolved along characteristics (in Lagrangianspirit) approximating advective form of equation

ft + afx = 0.

• Scheme II: convective schemethe solution is evolved over fixed point (in Eulerian spirit)approximating conservative form of equation

ft + (af )x = 0.

? a being a constant, with possible extension to a = a(x , t)(relativistic Vlasov equation).

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Scheme I: advective scheme

f n+1i = f (xi , t

n+1) = f (x?i , tn) = f (xi − ξ0∆x , tn), ξ0 =

a∆t

∆x

When ξ0 ∈ [0, 12 ]

u r r r u u u u u r r r u tn+1

u r r r u u

u u u r r r u tn

x0 xi−2 xi−1 xi xi+1 xi+2 xNξ0 ∈ [0, 1

2]

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Third order example

f n+1i = f n

i + (−1

6f ni−2 + f n

i−1 −1

2f ni −

1

3f ni+1)ξ0

+ (1

2f ni−1 − f n

i +1

2f ni+1)ξ2

0

+ (1

6f ni−2 −

1

2f ni−1 +

1

2f ni −

1

6f ni+1)ξ3

0 , (6)

? Method 1: apply WENO interpolation on (6).? No mass conservation.

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Matrix vector form

f n+1i = f n

i + ξ0(f ni−2, f

ni−1, f

ni , f

ni+1) · AL

3 · (1, ξ0, ξ20)′, (7)

with matrix

AL3 =

−1/6 0 1/6

1 1/2 −1/2−1/2 −1 1/2−1/3 1/2 −1/6

=

−1/6 0 1/65/6 1/2 −1/31/3 −1/2 1/6

0 0 0

−

0 0 0−1/6 0 1/65/6 1/2 −1/31/3 −1/2 1/6

.

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Conservative formulation

Rewrite the advective scheme into a conservative form

f n+1i = f n

i − ξ0((f ni−1, f

ni , f

ni+1) · CL

3 · (1, ξ0, ξ20)′

−(f ni−2, f

ni−1, f

ni ) · CL

3 · (1, ξ0, ξ20)′)

= f ni − ξ0(f n

i+1/2 − f ni−1/2)

with

CL3 =

−16 0 1

656

12 −1

313 −1

216

.

? Method 2: apply WENO reconstructions on flux functionsf ni+1/2? The scheme is locally conservative. However, such splittingcan NOT be generalized to equations with variable coefficients.

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Scheme II: conservative scheme

Integrating the conservative form of PDE

ft + (af )x = 0,

over [tn, tn+1] at xi gives

f n+1i = f n

i − (

∫ tn+1

tn

af (x , τ)dτ)x |x=xi

.= f n

i −Fx |x=xi

?= f n

i −1

∆x(Fi+ 1

2− Fi− 1

2) +O(∆xk)

where F(x) =∫ tn+1

tn af (x , τ)dτ , and Fi± 12

are numerical fluxes.

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

F(x)

F(xi ) =

∫ tn+1

tn

af (xi , τ)dτ =

∫ xi

x?i

f (ξ, tn)dξ,

by applying the Divergence Theorem on∫

Ω(ft + (af )x)dx = 0.


u r r r u u u u u r r r u tn

x0 xi−2 xi−1 xi xi+1 xi+2 xN

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

F(x)

F(xi ) =

∫ tn+1

tn

af (xi , τ)dτ =

∫ xi

x?i

f (ξ, tn)dξ,

by applying the Divergence Theorem on∫

Ω(ft + (af )x)dx = 0.


u r r r u u u

Ω

x?i6

u u r r r u tn

x0 xi−2 xi−1 xi xi+1 xi+2 xN

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Fi± 12

Let H(x) to be a function s.t.

F(x) =1

∆x

∫ x+ ∆x2

x−∆x2

H(ξ)dξ

then

Fx |x=xi =1

∆x(H(xi+ 1

2)−H(xi− 1

2)).

Let numerical fluxes Fi± 12

= H(xi± 12).

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

A first order scheme

f n+1i = f n

i − (

∫ tn+1

tn

af (x , τ)dτ)x

.= f n

i −Fx

= f ni −

1

∆x(F(xi )−F(xi−1)), if a > 0

= f ni −

1

∆x(

∫ xi

x?i

f (ξ, tn)dξ −∫ xi−1

x?i−1

f (ξ, tn)dξ)

Remark. When a is a constant and cfl < 1, the schemereduces to

f n+1i = f n

i −a∆t

∆x(f n

i − f ni−1)

which is the first order upwind scheme.

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

High order WENO reconstructions

High order WENO reconstructions are applied in the followingreconstructions

f (xi , tn)Ni=1

WENO−→

∫ xi

x?i

f (ξ, tn)dξ

N

i=1

WENO−→Fi+ 1

2

N

i=1

• Computational expensive: two weno reconstructionprocedures

• The reconstruction stencil is widely spread (not compact),leading to instability of algorithm.

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

f (xi , tn)Ni=1

WENO−→

∫ xi

x?i

f (ξ, tn)dξ

N

i=1

WENO−→Fi+ 1

2

N

i=1

f (xi , tn)Ni=1

WENO/ENO

−−−−−−−−−−−−−−−−−−−−−−→Fi+ 1

2

N

i=1

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Third order WENO reconstructionwhen a = 1, dt < dx .

Consider two substencils

S1 = xi−1, xi, S2 = xi , xi+1.

Let ξ = dtdx , the third order reconstruction from the three point

stencil xi−1, xi , xi+1

Fi+ 12

= γ1F (1)

i+ 12

+ γ2F (2)

i+ 12

where the linear weights

γ1 =1− ξ

3, γ2 =

2 + ξ

3,

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

and

F (1)

i+ 12

= (−(3

2ξ +

1

2ξ2)f n

i + (1

2ξ +

1

2ξ2)f n

i−1)dx

F (2)

i+ 12

= ((−1

2ξ +

1

2ξ2)f n

i − (1

2ξ +

1

2ξ2)f n

i+1)dx

Idea of WENO: adjust the linear weighting γi to a nonlinearweighting wi , such that

• wi is very close to γi , in the region of smooth structures,

• wi weights little on a non-smooth sub-stencil.

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

• Smoothness indicator:

β1 = (f ni−1 − f n

i )2, β2 = (f ni − f n

i+1)2

• Nonlinear weights

w1 = γ1/(ε+ β1)2, w2 = γ2/(ε+ β2)2

• Normalized nonlinear weights wi :

w1 = w1/(w1 + w2), w2 = w2/(w1 + w2)

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Summary of scheme II

Method 3

1 Reconstruction numerical fluxes Fi± 12

from WENO/ENO

reconstruction of f ni Ni=1

2

f n+1i = f n

i −1

∆x(Fi+ 1

2− Fi− 1

2),

? Compare with the method of line approach (Runge Kuttatime discretization), the time integration of the PDE here isexact.? The scheme can be extended to accommodate extra largetime step evolution and to the case of variable coefficienta(x , t).

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Comparison

• Method 1 & 2:• Approximates the advective form of PDE• Frame of reference: Lagrangian• Method 2 is the conservative formulation of Method 1 for

equations with constant coefficients• Method 1 can be generalized to equations with variable

coefficients

• Method 3:• Approximates the conservative form of PDE• Frame of reference: Eulerian• conservative scheme• extends to equations with variable coefficients

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Numerical Simulations

Method 1, 2, 3 with fifth order WENO reconstruction

• Linear advection equation

• Rigid body rotation

• Vlasov Poisson system

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Linear transport: ut + ux + uy = 0

Table: Order of accuracy with u(x , y , t = 0) = sin(x + y) at T = 20.dt = 1.1dx = 1.1dy .

– method 1 method 2 method 3

mesh error order error order error order

20×20 6.03E-4 – 8.28E-4 – 7.94E-4 –

40×40 2.24E-5 4.75 2.62E-5 4.97 2.51E-5 4.98

60×60 3.10E-6 4.88 3.44E-6 5.00 3.29E-6 5.01

80×80 7.50E-7 4.93 8.16E-7 5.00 7.80E-7 5.00

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

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Summary

Rigid body rotation:ut − yux + xuy = 0

Figure: Initial profile and numerical solution of method 1 from thenumerical mesh 100× 100, dt = 1.1dx = 1.1dy at T = 12π

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary


Figure: Method 2 and 3 with the numerical mesh 100× 100,dt = 1.1dx = 1.1dy at T = 12π.

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



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Summary


Figure: Method 1 (left) and 3 (right).

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



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Simulationresults

Summary


Figure: Method 1 (left) and 3 (right).

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Landau damping

Consider the VP system with initial condition,

f (x , v , t = 0) =1√2π

(1 + αcos(kx))exp(−v 2

2), (8)

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Weak Landau damping:α = 0.01, k = 2

Figure: Time evolution of L2 norm of electric field.

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

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Summary

Weak Landau damping (cont.)

Figure: Time evolution of L1 (upper left) and L2 (upper right) norm,discrete kinetic energy (lower left), entropy (lower right).

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

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Summary

Strong Landau damping:α = 0.5, k = 2

Figure: Time evolution of the L2 norm of the electric field.

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

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Summary

Strong Landau damping (cont.)

Figure: Time evolution of L1 (upper left) and L2 (upper right) norm,discrete kinetic energy (lower left), entropy (lower right).

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Two-stream instability

Consider the symmetric two stream instability, the VP systemwith initial condition

f (x , v , t = 0) =2

7√

2π(1 + 5v 2)(1 + α((cos(2kx)

+cos(3kx))/1.2 + cos(kx))exp(−v 2

2),

with α = 0.01, k = 0.5 and the length of domain inx−direction L = 2π

k .

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Figure: Two stream instability T = 53. The numerical solution ofmethod 1 (left) and Method 3 (right) with the numerical mesh64× 128.

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Figure: Third order semi-Lagrangian WENO method. Two streaminstability T = 53. The numerical mesh is 64× 128 (left) and256× 512 (right).

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Figure: Time development of L1 (upper left) and L2 (upper right)norm, discrete kinetic energy (lower left), entropy (lower right).

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

Ongoing and future work

• Algorithm:• Equations of variable coefficients• Truely multi-dimensional formulation of semi-Lagrangian

scheme evolving point values.

• Application• advection in incompressible flow• relativistic Vlasov equations

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kineticEquations

Jingmei Qiu

Introduction

Proposedmethod



Comparison

Simulationresults

Summary

THANK YOU!

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Semi-Lagrangian Formulations for Linear Advection ... · Equations Jingmei Qiu Introduction...

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