Numerical methods for reduced Vlasov equation

Model reduction Numerical methods and objectives Numerical results Perspectives

Numerical methods for reduced Vlasov equation

P. Helluy, L. Navoret, N. Pham

Inria Tonus & IRMA University of Strasbourg

NUMKIN 2016, IRMA, Octobre 20th 2016

P. Helluy, L. Navoret, N. Pham Vlasov reduced model NUMKIN 2016 1 / 25


Plan

1 Model reduction

2 Numerical methods and objectives

3 Numerical results

4 Perspectives



Model reduction of the Vlasov-Poisson equation

Vlasov equation :

∂f

∂t(x,v, t) + v · ∇xf(x,v, t) + E · ∇vf(x,v, t) = 0. (1)

f(x,v, t) : distribution function at time t ≥ 0, with positionx = (x1, x2, x3) ∈ Ωx ⊂ R3 and velocity v ∈ Ωv ⊂ R3

E(x, t) : electric field which satisfies the Poisson equation

E = −∇xΦ, with −4xΦ = ρ− 1. (2)

the charge density

ρ(x, t) =

ˆΩv

f(x,v, t) dv, (3)



Boundary condition

In practice Ωx =]0, L[3, with L > 0 and Ωv =]− Vmax, Vmax[3, whereVmax > 0 is the maximal velocity

Boundary condition

Periodical boundary condition in space :

Outflow boundary condition in velocity :

(E(x, t) · nv)− f(x,v, t) = 0

where nv be the outward normal unit vector on ∂Ωv



Reduced Vlasov equation

Weak formulation : Find f ∈ V =f ∈ L2(Ωv),E · ∇vf ∈ L2(Ωv)

∂t

ˆΩv

fϕ+∇x ·ˆ

Ωv

vfϕ+E ·ˆ

Ωv

∇vfϕ−βˆ∂Ωv

(E ·nv)−fϕ = 0, (4)

with any test function ϕ ∈ V and where β is a positive constant.The distribution function is approximated by

f(x,v, t) ≈Nv∑j=1

wj(x, t)ϕj(v). (5)

We obtain the reduced Vlasov equation

M∂tw +Ak∂xkw +B(E)w = 0, k = 1 . . . 3. (6)





M∂tw +Ak∂xkw +B(E)w = 0, k = 1 . . . 3.

In which w = (w1, w2, ..., wNv)T . The mass matrix M, Ak, k = 1 . . . 3 andB(E) are matrices of dimension Nv ×Nv, whose elements are given by

Mij =

ˆΩv

ϕiϕj , Akij =

ˆΩv

vk ϕiϕj , k = 1 . . . 3, (7)

B(E)ij =

ˆΩv

ϕi(E · ∇v)ϕj − βˆ∂Ωv

(E · nv)−ϕjϕi. (8)




Properties

The system depends only on x variable and not on the (x,v) variables

M is symmetric positive-definite, Ak is symmetric, k = 1, 2, 3, B(E)is “almost” skew-symmetric (except on boundaries)

Hyperbolic system

For 1/2 6 β 6 1 , the reduced Vlasov equation is L2 stable

1

2

d

dt

(ˆΩx×Ωv

f2

)6 −1

2

ˆΩx×∂Ωv

(E · nv)+ f2, (9)

Loss of total charge

d

dtρtot = −

ˆΩx×∂Ωv

(E · nv)+f, (10)



Plan

1 Model reduction


3 Numerical results

4 Perspectives



Numerical methods and objectives

Resolution of the hyperbolic system :

Finite volume (SeLaLib)

Semi-Lagrangian (SeLaLib)

Discontinous Galerkin (CLAC, SCHNAPS)

Objectives :

Implementation of the three methods

Comparison between the three methods



Finite volume scheme (2D)

Reduced Vlasov equation in 2D

∂tw +M−1A1∂x1w +M−1A2∂x2w +M−1B(E)w = 0

Finite-volume approximation

∂twkl =−F(wk,l,wk+1,l, ν

1)−F(wk−1,l,wk,l, ν1)

h1

−F(wk,l,wk,l+1, ν

2)−F(wk,l−1,wk,l, ν2)

h2+ S(wkl)

ν1 = (1, 0) and ν2 = (0, 1)

Viscous flux

F(wL,wR, ν) =M−1AiνiwL + wR

2−κ (wR −wL)

2, with κ ≥ 0.

(11)where κ : numerical diffusion.



Finite volume scheme (2D) - Stability

Time discretization : Runge-Kutta (RK) of order 1, 2, 3 ou 4.

1 For the centered flux

RK1 and RK2 schemes are unstable

RK3 and RK4 schemes : there exists a constant ℘ ∈ (0, 1) such thatthe schemes are stable under the CFL condition ∆t = ℘ ∆x

Vmax.

2 For the viscous flux : The Runge-Kutta schemes of order 1, 2, 3, 4with the viscous flux are all stable when taking ∆t sufficiently small(CFL condition).

RK1 and RK2 : CFL parameter depends on κ

RK3 and RK4 schemes : CFL parameter does not depend on κ



Finite volume scheme (2D) - Implementation

Implementation

SeLaLib libraby

MPI parallelization

Drawbacks

Matrix inversion (skyline storage of band matrices)

stability under CFL condition (in space and velocity)

low spatial accuracy

Advantages

locally conservative (mass, momentum) and L2 stable

finite-element refinement in the velocity direction

numerical dissipation only on the spatial direction



Semi-Lagrangian scheme (2D)

Lagrange polynomials and Gauss-Lobatto points ⇒ diagonal matrices

M−1Ai =

vi1. . .

viNv

⇒ Semi-Lagrangian approximation

Splitting method

Advection in the x1 direction (1D Semi-Lagrangian)Advection in the x2 direction (1D Semi-Lagrangian)Addition of the source term

Implementation :

Selalib library

MPI parallelization

Advantages :

No CFL condition (in space)

High order method

Drawbacks :

Mass conservative on uniform 1D mesh

difficult to extend to non-cartesian meshP. Helluy, L. Navoret, N. Pham Vlasov reduced model NUMKIN 2016 13 / 25


Discontinuous Galerkin scheme - Drift-kinetic model

In a cylinder consider a strong homogeneous magnetic field Bapp = b e3,the Vlasov equation

∂f

∂t+ v · ∇xf + (E + v ×Bapp) · ∇vf = 0.

becomes the drift-kinetic equation

∂tf + E⊥⊥ · ∇x⊥f + v||∂x||f + E||∂v||f = 0. (12)

Weak formulation (in velocity) of drift-kinetic equation

M∂tw + E⊥ · M∇⊥w +A∂x||w +B(Ex||)w = 0. (13)



Discontinuous Galerkin scheme - Drift-kinetic model

Weak upwind DG formulation

ˆL∂tw

L · ψL −ˆLE⊥w

L · ∇⊥ψL +

ˆ∂L∩Ωx

E⊥(n+wL + n−wR

)· ψL+

ˆLwL · (M−1A)∂3ψ

L +

ˆ∂L∩Ωx

((M−1A)n+3 wL + (M−1A)n−3 wR) · ψL+

ˆL

(M−1B(EX3))wL · ψL = 0

(14)

where n the normal vector on ∂L oriented from the cell L to the

neighboring cells R, ψL,k : test function.



Discontinuous Galerkin scheme

Implementation :

SCHNAPS code

multipatch, curved hexahedronmesh (gmsh)

GPU parallelization (OpenCL)

Starpu (task distribution)

Figure: Macro-mesh of a simple disk

Drawbacks :

CFL condition (in velocity andspace)

Advantages :

high-order accuracy

handle complex geometry

parallelization



Plan

1 Model reduction


3 Numerical results

4 Perspectives



2D Landau damping

Initial distribution function : f0(x, v) =12π

(1 + ε cos(k1x1) cos(k2x2))e−

(v21+v2

2)

2 .

Parameters : k1 = k2 = 0.5, ε = 5× 10−3 ; x ∈ [0, 4π]× [0, 4π], v ∈]− 6, 6[×]− 6, 6[ ;N1 = N2 = 32 (space), NC1 = NC2 = 32, degree 2 (velocity) ; 4 processors.

∆t cpu time (s)FV 1.47 10−2 10182SL 0.1 748SL 0.5 171



Speed-up

2D Landau damping : Computed on the supercomputer Curie (TGCC).Parameters : 100 iterations, 2562 × 652 ≈ 277.106 unknows

Number of CPU 4 8 16 32 64 128

computation time 21280 13992 7237 3685 1869 985

speed-up (relative) 1.52 1.93 1.96 1.97 1.89

speed-up 6.08 11.76 23.1 45.53 86.41

Table: Computational time and speedup for the 2D finite-volume code.

Number of processor units (CPU) 8 16 64 128

computation time (second) 7398 3673 943 491

speed-up (relative) 2.01 3.89 1.92

speed-up 16.11 62.76 120.54

Table: Computational time and speedup for the 2D semi-Lagrangian code.



Efficiency of code

The efficiency defined by

eff :=s ∗ nbpoints ∗ nbiterationsT ∗ 106 ∗ nbprocessors

. (15)

With the finite volume code, eff = 1.75

With the semi-Lagrangian code, eff = 3.5.

→ The effeciency of semi-Lagrangian method is twice as much than thisof the finite volumes method. Reason : semi-Lagrangian is withoutmatrices inversions.



Guiding center model

Guiding center model in a disc

∂tρ+ E⊥ · ∇xρ = 0

−∆xΦ = ρ− ρ, E = −∇xΦ

Diocotron instability test-case [Davidson, 2013] : perturbed annularelectron layer

Growth-rate of the 4th Fourier modeρ at time t = 100



Speed-up

Computational time and speedup for the 2D schnaps code on CPU.Computed on the Irma Atlas cluster. Parameters : 20 iterations, Mx = 20dx = 3, Nv = 31, dv = 3,Mv = 20.

sequential code OpenCL code OpenCL code1 core 1 core 1GPU

20 iterations 3532 2401.3 39.2

1 iterations 168.19 114.34 1.8

relative speed-up 1.47 63.5

Nb of CPU 1 2 4 8 16 32 64

comp. time 2401.3 1396.12 801.39 504.6 276.89 193.9 142.5

speed-up 1.72 2.997 4.759 8.67 12.38 16.9



Plan

1 Model reduction


3 Numerical results

4 Perspectives



Perspectives

1 Apply the code in schnaps to solve the drift-kinetic model and thegyro-kinetic model.

2 Resolve the drift-kinetic in SELALIB to compare with the one inschnaps.

3 Study the Vlasov model with the collision term.



THANKS FOR YOURATTENTION !


Numerical methods for reduced Vlasov equation

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