A high-order positivity preserving method for the Vlasov ......method for the Vlasov-Poisson system...

Issues in Solving the Boltzmann Equation forAerospace ApplicationsICERM, Providence, RI

A high-order positivity preservingmethod for the Vlasov-Poisson system

James A. Rossmanith 1, David C. Seal 2,Andrew J. Christlieb 2

1Iowa State University2Michigan State University

June 7th, 2013

Outline

Introduction

Hybrid discontinous-Galerkin method

Examples

Other Extensions & Conclusions

Vlasov-Poisson System: A Kinetic Plasma ModelWhat is a plasma?

I ‘Fourth’ state of matter, and most abundant form of ordinary matter.Given enough energy, anything will turn into plasma.

I Can think of plasma as an ionized gas or liquid. Therefore modelequations must respect E&M forces plus liquid/gas dynamics.

Scientific and Engineering Applications of Plasma

I Astrophysics (stars, solar corona/wind)I Nuclear experiments, tocamaks, stelleratorsI Lightning/Polar AuroraeI Flourescent lights, TVs, etc.

Mathematical ModelsI Fluid Models (e.g. continuum equations in gas dynamics)I Kinetic Models - Computationally expensive.

Vlasov-Poisson System: A Collisionless Plasma

I Probability distribution function (PDF):

f(t,x,v) := probability of finding an electron at t & x, w/ vel. v

I Boltzmann equation (non-dimensional units):

∂f

∂t+ v · ∇xf + a · ∇vf = C(f)

Two species Vlasov-Poisson (Electrostics + C(f) ≡ 0)

a = −∇φ, −∇2φ = ρ(t,x)− ρ0.

I Some numerical challenges

1. High dimensionality (3 + 3 + 1) - even worse for Boltzmann!2. Conservation of mass & total energy, positivity3. Complex geometries, boundary conditions.4. Small time steps due to v ∈ R3 or strong electric fields.

Eulerian schemesSelecting a scheme for Vlasov-Poisson

I Popular “finite-X” methods:

1. Finite-Difference2. Finite-Volume3. Finite-Element4. Discontinuous-Galerkin (high-order + unstructured grids)

I Main advantage:1. Fast convergence2. Provable accuracy, load balancing for parallel computing.

I Main disadvantage:

1. Curse of dimensionality2. Courrant-Fredricks-Lewy (CFL) limits

I M th-order discontinuous-Galerkin: ν ≈ 1/ (2M − 1).3. Diffusion (in velocity space) - bad!

I Some recent results for Vlasov-Poisson:[Banks & Hittinger, ’10],[Heath, Gamba, Morrison, Michler ’11],[Cheng, Gamba, Morrison ’13], . . .

Particle-in-cell and semi-Lagrangian methodsSelecting a scheme for Vlasov-Poisson

1. Particle in Cell 2. semi-Lagrangian1.

I Discretize f with “macro-particles.”I Key difficulties: statistical noise ∼ O(1/

√N).

I [Birdsall & Langdon, 1985], [Hockney & Eastwood, 1989]2.

I Main advantage: Remove CFL constraint, maintain mesh rep., . . .I Main disadvantage: Boundary conditions and unstructured grids?

Mesh distortions from Lagrangian evolution + interpolation error.I [Parker & Hitchon, ’97], [Sonnendrücker et al. ’99],

[Güçlü & Hitchon, ’12], . . .

Outline

Introduction


Examples


Splitting Techniques for Vlasov-Poisson

Semi-Lagrangian w/ Strang splitting for Vlasov:[Cheng & Knorr, 1976], [Besse & Sonnendrücker, ’03],[Qiu & Christlieb, ’09], . . .Split f,t + v · f,x +∇φ(t,x) · f,v = 0 into:

1. ∆t2

step on: f,t + v · f,x = 0

2. Solve -∇2φ = ρn+12 − ρ0 and compute En+

12 = −∇φ.

3. ∆t step on: f,t + En+12 · f,v = 0.

4. ∆t2

step on: f,t + v · f,x = 0

High-order Runge-Kutta Nyström splitting for V-P is an option.[Rossmanith & S, ’10], [Crouseilles et al, ’11], [S, ’12]

Multi-D extensions of SLDG?I Natural extension =⇒ Cartesian.I Hybrid = Which method do we want to apply to sub-problems?

Hybrid SLDG (HSLDG) for Vlasov-Poisson

Proposed hybrid DG scheme

1. SLDG (Semi-Lagrangian DG) on velocity space: f,t +En+12 · f,v = 0.

[Rossmanith & S, ’10], [Einkemmer & Ostermann, ’12],not [Restelli et al, ’06], [Qiu & Shu, ’11] (less efficient).

I Removes CFL condition.

2. RKDG (Runge-Kutta DG) on configuration space: f,t + v · f,x = 0.[Reed & Hill, 1976], [Cockburn & Shu, 90’s]

I Unstructured grids and Boundary conditions.I Sub-cycle independent problems.

I Disclosure: SLDG better for structured grids!

Building blocksNeed to define how sub-problems are tackled.

Semi-Lagrangian DG (SLDG)

Advection Equation: f,t + a f,v = 0

1. Reconstruct. The original projection:

F(k),ni :=

1

∆v

∫Ciϕ

(k)i (v)f(t

n, v) dv; f(t, v) =∑i,k

F(k)i (t)ϕ

(k)i (v).

2. Evolve. The evolution step is simply the exact solution:

f(t, v) = f0(v − at); f0(v) = f(0, v).

3. Average. This step requires the solution from step 2.

F(k),n+1i :=

1

∆v

∫Ciϕ

(k)i (v)f(t

n+1, v) dv

=1

∆v

∫Ciϕ

(k)i (v)f(t

n, v − a(t− tn))) dv

2D Semi-Lagrangian DGConstant coefficient advection: a.k.a. corner transport + high-order

1. Forward (cell discontinuities) 2. Backward (quadrature points)

Advection equation: f,t − E1 f,vx − E2f,vy = 0

F(k),n+1i,j :=

1∣∣Ti,j∣∣∫Ti,j

ϕ(k)i,j (x, y) f

h (tn+1, x, v) dx dv=

1∣∣Ti,j∣∣4∑

m=1

∫Tmi,j

ϕ(k)i,j (x, y)f

h (tn, x+ ∆tE1, v + ∆tE2) dx dvI Mass conservation (and stability!) come from exact integration

Hybrid DG: Reduction to sub-problemsNot trivial because basis functions depend on transverse variable

1. Reconstruct: Start with a full 2N -dimensional solution:

fh (tn,x,v)∣∣∣T 2Ni

=∑k2

F(k2)i (t

n)ϕ(k2)2N (x,v)

2. Project full solution onto to sub-problems (N -dimensional) by takingslices at quadrature points {v1,v2, . . .vMN }:

fh (tn,x,vm)∣∣∣TNi

=∑k

F(k)N,i(t

n)ϕ(k)N (x)

3. Evolve (RKDG or SLDG) sub-problems:

fh(tn+1,x,vm

)∣∣∣TNi

=∑k

F(k)N,i(t

n+1)ϕ(k)N (x)

4. Integrate: Sub-problems integrated up to 2N -dimensional problem:

fh(tn+1,x,v

)∣∣∣T 2Ni

=∑k2

F(k2)i (t

n+1)ϕ(k2)2N (x,v)

1D-1V Example: f,t + v f,x = 0

1. Full 2D-solution. 2. 1D-Problems (at quadrature points).

3. Evolve lower-D problems (SLDC/RKDG) 4. Integrate up to full soln.

HSLDG for Vlasov-Poisson: Sub-Cycling

Basic Idea: March sub-problems at their own pace

I Many independent sub-Problems: f,t + vmf,x = 0. Global restriction:

∆t ≤ CFL ∆x|vmax|� CFL ∆x|vm|

∼ ∆tlocal

I For each equation, define: ∆tlocal := ∆t/max{

1,⌈|vm|∆tν∆x

⌉}.

Outline

Introduction


Examples


Example: Forced Vlasov-Poisson Test ProblemI Method of Manufactured Solutions produces a source term:

f,t + vf,x + E(t, x)f,v = ψ(t, x, v), E,x =

∫ ∞−∞

f(t, x, v) dv −√π.

I Choose the exact solution: f(t, x, v) = {2− cos (2x− 2πt)} e−14

(4v−1)2 .

I New Hybrid Operators:

Sub-cycled RKDG on Problem A: f,t + v f,x = ψ(t, x, v),SLDG on Problem B: f,t + E(t, x) f,v = 0,

Mesh HSL2 Error log2(Ratio) HSL4 Error log2(Ratio)102 5.193× 10−1 – 1.085× 100 –202 1.432× 10−1 1.858 2.725× 10−1 1.993402 1.640× 10−2 3.126 1.652× 10−2 4.044802 3.438× 10−3 2.255 7.058× 10−4 4.5481602 8.333× 10−4 2.045 3.421× 10−5 4.3673202 2.068× 10−4 2.011 1.953× 10−6 4.1316402 5.161× 10−5 2.003 1.197× 10−7 4.028

Example: Landau Damping with HSLDG

Initial Conditions:

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

1 + α cos(kx)) 1√

2πe−

v2

2

Weak Landau Strong Landau

0 10 20 30 40 50 6010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

0 10 20 30 40 50 6010

−4

10−3

10−2

10−1

100

101

α = 0.01 α = 0.5

Plasma Sheath Simulation (1D-1V) with HSLDGInitial conditions (in dimensional quantitites):

f̃(0, x̃, ṽ) = f̃0(ṽ) =ρ̃0√

2πmkθ̃0exp

(−mṽ

2

2kθ̃0

), x̃ ∈ [0, L].

Boundary conditions:

f̃(t, x̃ = 0, ṽ) = 0, f̃(t, x̃ = L, ṽ) = f̃0(ṽ),

φ̃(t, x̃ = 0) = 0, φ̃x̃(t, x̃ = L) = 0.

1 eV electrons, 0.1m domain, density ρ̃0 = 9.10938188× 10−18 kgm .

Plasma Sheath Simulation (1D-1V) with HSLDG

(sheath-movie.mp4)

Lavf52.93.0

sheath-movie.mp4Media File (video/mp4)


0 0.02 0.04 0.06 0.08 0.10

2.5

5

7.5

10

12.5x 10

12n

e(t,x) at t = 3.0269e−06

0 0.02 0.04 0.06 0.08 0.1−800

−600

−400

−200

0

200

400

600

800

E(t,x) at t = 3.0269e−06

Number density Electric field


I Initial conditions: constant density, ρ̃0 = 9.10938188× 10−18 kgm2 .

I Boundary conditions: f̃(t̃, ‖x̃‖ = L, ṽ) = 0, φ̃(t̃, ‖x̃‖ = L) = 0.

−0.1 −0.05 0 0.05 0.1−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

rho(t,x,y) at t = 0 [DoGPack]

0 0.02 0.04 0.06 0.08 0.10

2.5

5

7.5

10

12.5x 10

12n

e(t,r) at t = 0 [DoGPack]

1 eV electrons,(π · 0.12

)m2 domain.


−0.1 −0.05 0 0.05 0.1−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

rho(t,x,y) at t = 4.4843e−08 [DoGPack]

0 0.02 0.04 0.06 0.08 0.10

2.5

5

7.5

10

12.5x 10

12n

e(t,r) at t = 4.4843e−08 [DoGPack]


−0.1 −0.05 0 0.05 0.1−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1


0 0.02 0.04 0.06 0.08 0.10

2.5

5

7.5

10

12.5x 10

12n


Plasma Sheath Simulation (2D-2V) with HSLDGCross sections of velocity space

−0.2 −0.1 0 0.1 0.2

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

f(t,vx,vy) at t = 0 [DoGPack]

0

50

100

150

200

250

300

−0.2 −0.1 0 0.1 0.2

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2


−25

−20

−15

−10

−5

0

5

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15

20

25

Plasma Sheath Simulation (2D-2V) with HSLDGCross section of velocity space

−0.2 −0.1 0 0.1 0.2

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2


−25

−20

−15

−10

−5

0

5

10

15

20

25

−0.2 −0.1 0 0.1 0.2

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

f(t,vx,vy) at t = 5.6054e−08 [DoGPack]

−25

−20

−15

−10

−5

0

5

10

15

20

25

−0.2 −0.1 0 0.1 0.2

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2


−25

−20

−15

−10

−5

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25

−0.2 −0.1 0 0.1 0.2

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2


−25

−20

−15

−10

−5

0

5

10

15

20

25

Outline

Introduction


Examples


Conclusions & Future Work

I Vlasov-Poisson: nonlinear/nonlocal advection equation

I Semi-Lagrangian and Eulerian grid-based methods as alternativeto PIC and particle methods

I Operator splitting + sub-cycling + semi-Lagrangian relaxes CFL.

I 1D-1V and preliminary 2D-2V results: collisionless simulation ofthe formation of a plasma sheath, Landau damping, bump ontail, ... http://www.dogpack-code.org

I The Hybrid HSLDG and the SLDG Method attain:

1. Unconditionally stable; (sub-cycling / semi-Lagrangian)2. High-order space (5th order) and time (4th order);3. Mass conservative; and4. Positivity-preserving

Future workI Extra options for 2D-Poisson solver.

I Vlasov-Maxwell, SLDG on even higher dimensions, . . .

Conclusions

Thank You!

This work was partially supported by the following agencies:I Michigan State University Foundation SPG-RG100059I Air Force Office of Scientific Research (AFOSR)

FA9550-11-1-0281, FA9550-12-1-0343, FA9550-12-1-0455I National Science Foundation (NSF) DMS-1115709

The DoGPack software packagehttp://www.dogpack-code.org

Semi-Lagrangian discontinuous GalerkinPositivity Preserving Limiter: Step I

Provided cell average is positive, one can gaurantee as many points asrequested are positive without losing order of accuracy.1. To limit cell Tij , compute

m := min(x,v)∈Tij

fh(x, v),

or at least an approximate minimum.

2. Define the limited solution [Zhang & Shu, 2010] as

f̃h(ξ, η) := F (1) + θ ·(fh(ξ, η)− F (1)

),

θ = min

{1,

F (1)

F (1) −m

}, 0 ≤ θ ≤ 1.

3. Main advantages: Locally applied limiter, easy to implement.

4. Difficulties:

I Exact minimum of the function on each cell?I How does one make sure average is positive?

Semi-Lagrangian discontinuous GalerkinPositivity Preserving Limiter [Rossmanith and S., 2011]: Step II

Theorem:A single advect-project step preserves positivity in the mean, and hence theoverall algorithm is positivity preserving. That is,[

F(1)i,j

]n+1:=

1∣∣Ti,j∣∣∫Ti,j

fh(tn+1, x, v

)dx dv ≥ 0.

Proof:I The points chosen for integration are forced to be positive via a

judicious and finite choice of limiting points in a preprocessing step:

I Step I makes solution positive at our choice of quadrature points.I All quadrature weights are positive.I Extra effort reduces number of points to 2MK, where K :=

⌈M2

⌉.

Semi-Lagrangian DG-FEM4th order-in-time

I Poisson’s equation−∇2φ = ρn − ρ0

I Time derivatives

En = ∇φ, En,t = − (ρu)n , En,tt = ∇ · En − (ρE)n

En,ttt = ∇ · (ρuE + ρEu)n −∇ · ∇ · Fn + En∇ · (ρu)n +(ρ2u

)nĒ(t,x) = En + (t− tn)En,t +

1

2(t− tn)2 En,tt +

1

6(t− tn)3 En,ttt

Fourth-order splitting:Stage 1: + 0.6756 ∆t step on f,t + v · f,x = 0Stage 2: + 1.3512 ∆t step on f,t + Ē (t,x) · f,v = 0Stage 3: − 0.1756 ∆t step on f,t + v · f,x = 0Stage 4: − 1.7024 ∆t step on f,t + Ē (t,x) · f,v = 0Stage 5: − 0.1756 ∆t step on f,t + v · f,x = 0Stage 6: + 1.3512 ∆t step on f,t + Ē (t,x) · f,v = 0Stage 7: + 0.6756 ∆t step on f,t + v · f,x = 0

Eulerian Discontinuous Galerkin methods

Spatial discretization [Cockburn & Shu, 1990’s]I Basis functions: ϕ(`)(x) =

{1, x, y, x2, xy, y2, . . .

}I Galerkin expansion: fh(t,x) =

∑M(M+1)/2k=1 F

(`)(t)ϕ(`)(x)

I Semi-discrete weak-form of f,t +∇ ·R(f) = 0:∫Tϕ(`) f,t dx = −

∫Tϕ(`)∇ ·R(f) dx

=⇒ ddtF (`) =

1

|T |

∫T∇ϕ(`) ·R(f) dx︸︷︷︸Interior

− 1|T |

∮∂T

ϕ(`) R(f) · ds︸︷︷︸Edge

I Interior: quadrature, Edge: quadrature, then approx Riemann soln

IntroductionHybrid discontinous-Galerkin methodExamplesOther Extensions & Conclusions

A high-order positivity preserving method for the Vlasov ......method for the Vlasov-Poisson system...

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Transcript of A high-order positivity preserving method for the Vlasov ......method for the Vlasov-Poisson system...