Issues in Solving the Boltzmann Equation for Aerospace Applications ICERM, Providence, RI A high-order positivity preserving method for the Vlasov-Poisson system James A. Rossmanith 1 , David C. Seal 2 , Andrew J. Christlieb 2 1 Iowa State University 2 Michigan State University June 7 th , 2013
Outline Introduction Hybrid discontinous-Galerkin method Examples Other Extensions & Conclusions
Outline Introduction Hybrid discontinous-Galerkin method Examples Other Extensions & Conclusions
Vlasov-Poisson System: A Kinetic Plasma Model What is a plasma? ◮ ‘Fourth’ state of matter, and most abundant form of ordinary matter. Given enough energy, anything will turn into plasma. ◮ Can think of plasma as an ionized gas or liquid. Therefore model equations must respect E&M forces plus liquid/gas dynamics. Scientific and Engineering Applications of Plasma ◮ Astrophysics (stars, solar corona/wind) ◮ Nuclear experiments, tocamaks, stellerators ◮ Lightning/Polar Aurorae ◮ Flourescent lights, TVs, etc. Mathematical Models ◮ Fluid Models (e.g. continuum equations in gas dynamics) ◮ Kinetic Models - Computationally expensive.
Vlasov-Poisson System: A Collisionless Plasma ◮ Probability distribution function (PDF): f ( t, x , v ) := probability of finding an electron at t & x , w/ vel. v ◮ Boltzmann equation (non-dimensional units): ∂f ∂t + v · ∇ x f + a · ∇ v f = C ( f ) Two species Vlasov-Poisson (Electrostics + C ( f ) ≡ 0 ) −∇ 2 φ = ρ ( t, x ) − ρ 0 . a = −∇ φ, ◮ Some numerical challenges 1. High dimensionality ( 3 + 3 + 1 ) - even worse for Boltzmann! 2. Conservation of mass & total energy, positivity 3. Complex geometries, boundary conditions. 4. Small time steps due to v ∈ R 3 or strong electric fields.
Eulerian schemes Selecting a scheme for Vlasov-Poisson ◮ Popular “finite-X” methods: 1. Finite-Difference 2. Finite-Volume 3. Finite-Element 4. Discontinuous-Galerkin (high-order + unstructured grids) ◮ Main advantage: 1. Fast convergence 2. Provable accuracy, load balancing for parallel computing. ◮ Main disadvantage: 1. Curse of dimensionality 2. Courrant-Fredricks-Lewy (CFL) limits ◮ M th -order discontinuous-Galerkin: ν ≈ 1 / (2 M − 1) . 3. Diffusion (in velocity space) - bad! ◮ Some recent results for Vlasov-Poisson: [Banks & Hittinger, ’10] , [ Heath, Gamba, Morrison, Michler ’11 ], [ Cheng, Gamba, Morrison ’13 ], . . .
Particle-in-cell and semi-Lagrangian methods Selecting a scheme for Vlasov-Poisson 1. Particle in Cell 2. semi-Lagrangian 1. ◮ Discretize f with “macro-particles.” √ ◮ Key difficulties: statistical noise ∼ O (1 / N ) . ◮ [Birdsall & Langdon, 1985] , [Hockney & Eastwood, 1989] 2. ◮ Main advantage: Remove CFL constraint, maintain mesh rep., . . . ◮ Main disadvantage: Boundary conditions and unstructured grids? Mesh distortions from Lagrangian evolution + interpolation error. ◮ [Parker & Hitchon, ’97] , [Sonnendrücker et al. ’99] , [Güçlü & Hitchon, ’12] , . . .
Outline Introduction Hybrid discontinous-Galerkin method Examples Other Extensions & Conclusions
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: ∆ t 1. 2 step on: f ,t + v · f , x = 0 - ∇ 2 φ = ρ n + 1 E n + 1 2 − ρ 0 2 = −∇ φ . 2. Solve and compute 3. ∆ t step on: f ,t + E n + 1 2 · f , v = 0 . ∆ t 4. 2 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? ◮ Natural extension = ⇒ Cartesian. ◮ 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 + E n + 1 2 · f , v = 0 . [ Rossmanith & S, ’10 ], [ Einkemmer & Ostermann, ’12 ], not [ Restelli et al, ’06 ], [ Qiu & Shu, ’11 ] (less efficient). ◮ Removes CFL condition. 2. RKDG (Runge-Kutta DG) on configuration space: f ,t + v · f , x = 0 . [ Reed & Hill, 1976 ], [ Cockburn & Shu, 90’s ] ◮ Unstructured grids and Boundary conditions. ◮ Sub-cycle independent problems. ◮ Disclosure: SLDG better for structured grids! Building blocks Need to define how sub-problems are tackled.
Semi-Lagrangian DG (SLDG) Advection Equation: f ,t + a f ,v = 0 1. Reconstruct. The original projection: � 1 � F ( k ) ,n ϕ ( k ) F ( k ) ( t ) ϕ ( k ) ( v ) f ( t n , v ) dv ; := f ( t, v ) = ( v ) . i i i i ∆ v C i i,k 2. Evolve. The evolution step is simply the exact solution: f ( t, v ) = f 0 ( v − at ); f 0 ( v ) = f (0 , v ) . 3. Average. This step requires the solution from step 2. � 1 F ( k ) ,n +1 ϕ ( k ) ( v ) f ( t n +1 , v ) dv := i i ∆ v C i � 1 ϕ ( k ) ( v ) f ( t n , v − a ( t − t n ))) dv = i ∆ v C i
2D Semi-Lagrangian DG Constant coefficient advection: a.k.a. corner transport + high-order 1. Forward (cell discontinuities) 2. Backward (quadrature points) Advection equation: f ,t − E 1 f ,vx − E 2 f ,vy = 0 � 1 i,j ( x, y ) f h � � F ( k ) ,n +1 ϕ ( k ) t n +1 , x, v := dx dv � � i,j � T i,j � T i,j 4 � 1 � i,j ( x, y ) f h � t n , x + ∆ tE 1 , v + ∆ tE 2 � ϕ ( k ) = dx dv � � � T i,j � T m m =1 i,j ◮ Mass conservation (and stability!) come from exact integration
Hybrid DG: Reduction to sub-problems Not trivial because basis functions depend on transverse variable 1. Reconstruct: Start with a full 2 N -dimensional solution: � � f h ( t n , x , v ) F ( k 2 ) ( t n ) ϕ ( k 2 ) � = 2 N ( x , v ) � i T 2 N i k 2 2. Project full solution onto to sub-problems ( N -dimensional) by taking slices at quadrature points { v 1 , v 2 , . . . v M N } : � � f h ( t n , x , v m ) F ( k ) N,i ( t n ) ϕ ( k ) � = N ( x ) � T N i k 3. Evolve (RKDG or SLDG) sub-problems: �� � f h � t n +1 , x , v m F ( k ) N,i ( t n +1 ) ϕ ( k ) � = N ( x ) � T N i k 4. Integrate: Sub-problems integrated up to 2 N -dimensional problem: �� � f h � F ( k 2 ) ( t n +1 ) ϕ ( k 2 ) t n +1 , x , v � = 2 N ( x , v ) � i T 2 N i k 2
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 ◮ Many independent sub-Problems: f ,t + v m f ,x = 0 . Global restriction: ∆ t ≤ CFL ∆ x | v max | ≪ CFL ∆ x | v m | ∼ ∆ t local � � �� | v m | ∆ t ◮ For each equation, define: ∆ t local := ∆ t/ max 1 , . ν ∆ x
Outline Introduction Hybrid discontinous-Galerkin method Examples Other Extensions & Conclusions
Example: Forced Vlasov-Poisson Test Problem ◮ Method of Manufactured Solutions produces a source term: � ∞ f ( t, x, v ) dv − √ π. f ,t + vf ,x + E ( t, x ) f ,v = ψ ( t, x, v ) , E ,x = −∞ ◮ Choose the exact solution: f ( t, x, v ) = { 2 − cos (2 x − 2 πt ) } e − 1 4 (4 v − 1) 2 . ◮ 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 log 2 (Ratio) HSL4 Error log 2 (Ratio) 10 2 5 . 193 × 10 − 1 1 . 085 × 10 0 – – 20 2 1 . 432 × 10 − 1 2 . 725 × 10 − 1 1 . 858 1 . 993 40 2 1 . 640 × 10 − 2 1 . 652 × 10 − 2 3 . 126 4 . 044 80 2 3 . 438 × 10 − 3 7 . 058 × 10 − 4 2 . 255 4 . 548 160 2 8 . 333 × 10 − 4 3 . 421 × 10 − 5 2 . 045 4 . 367 320 2 2 . 068 × 10 − 4 1 . 953 × 10 − 6 2 . 011 4 . 131 640 2 5 . 161 × 10 − 5 1 . 197 × 10 − 7 2 . 003 4 . 028
Example: Landau Damping with HSLDG Initial Conditions: � � 1 e − v 2 f ( t = 0 , x, v ) = 1 + α cos( kx ) √ 2 2 π Weak Landau Strong Landau −1 1 10 10 −2 10 0 10 −3 10 −1 10 −4 10 −5 10 −2 10 −6 10 −3 10 −7 10 −8 −4 10 10 0 10 20 30 40 50 60 0 10 20 30 40 50 60 α = 0 . 01 α = 0 . 5
Plasma Sheath Simulation (1D-1V) with HSLDG Initial conditions (in dimensional quantitites): � � v 2 ρ 0 ˜ − m ˜ ˜ v ) = ˜ f (0 , ˜ x, ˜ f 0 (˜ v ) = exp , ˜ x ∈ [0 , L ] . � 2 k ˜ 2 πmk ˜ θ 0 θ 0 Boundary conditions: ˜ ˜ v ) = ˜ f ( t, ˜ x = 0 , ˜ v ) = 0 , f ( t, ˜ x = L, ˜ f 0 (˜ v ) , ˜ ˜ φ ( t, ˜ x = 0) = 0 , φ ˜ x ( t, ˜ x = L ) = 0 . ρ 0 = 9 . 10938188 × 10 − 18 kg 1 eV electrons, 0.1m domain, density ˜ m .
Plasma Sheath Simulation (1D-1V) with HSLDG (sheath-movie.mp4)
Plasma Sheath Simulation (1D-1V) with HSLDG
Plasma Sheath Simulation (1D-1V) with HSLDG E(t,x) at t = 3.0269e−06 n e (t,x) at t = 3.0269e−06 12 12.5 x 10 800 600 10 400 200 7.5 0 5 −200 −400 2.5 −600 −800 0 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 Number density Electric field
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