A Discontinuous Galerkin Method for Vlasov Systems

Transcription

1 A Discontinuous Galerkin Method for Vlasov Systems P. J. Morrison Department of Physics and Institute for Fusion Studies The University of Texas at Austin morrison/ UBC, February 27, 2012 Collaborators: I. Gamba, Y. Cheng, F. Li, R. Heath, C. Michler, J-M. Qiu,...

2 Hot Magnetized Plasmas Hot, magnetized plasmas

3 Hot Magnetized Plasma Ionized gas of charged particles where Hot collisions not important Rare collisions i.e. when mean free path is very long Magnetized magnetic field important Gyroradius small compared to other scale lengths

4 Vlasov Equation: Maxwell-Vlasov System f α (x, v, t) t + v f α + e ( ) α E + v B v f α = f ) α m α t c 0 where f is phase space density, α = e, i is species index, and the sources, charge density and current density, are given by ρ(x, t) = α e α IR 3 d3 v f α, J(x, t) = α e α IR 3 d3 v vf α, which couple into Maxwell s Equations: B t ɛ 0 E t = E, B = 0 = B µ 0 J, ɛ 0 E = ρ

5 Vlasov Regularity Vlasov-Poisson: (1952) 3D stellar dynamics. R. Kurth local existence in time. (1977) spherical symmetry. J. Batt, global existence. (1989) 3D compact support. K. Pfaffelmoser, B. Perthame, J. Schaeffer, smooth global existence.... Maxwell-Vlasov: Open!

6 Maxwell- Vlasov Regularity R. Glassey, J. Schaeffer,... After 40 years we have precious little to show for it. Computation?

7 Tokamak 24 m Vacuum Vessel Magnet System Shield 30 m R=6.2 m I p =15 MA P fus =500 MW Divertor Person 1st June 2006 ITER Technical Overview - Ljubljana 8

8 Vlasov-Maxwell Multiscale Computation Range of scales Time scale (seconds) e 1 i L/v te L/v ti c Gyromotion Turbulence Mean profiles Collisions Space scale (meters) e i L Gyromotion Turbulence Mean Profiles Collisions

9 Expense (brute force) Time scale (seconds) Curse of Dimension Temporal grid: ~10 13 time steps Collisions Space scale (meters) Spatial grid: ~10 6 grid points x 3-D = grid points Velocity grid: ~10 grid points x 3-D = 10 3 grid points Total: ~10 34 total grid points Velocity grid: 100 grid points Total: total grid points

10 Feasibility petaflop=10 15 opers sec petaflop 10 6 in parallel opers sec opers sec π 107 sec year = 1028 opers year = 10 9 age of solar system < age of universe

11 Maxwell-Vlasov System (to scale) f α (x, v, t) t + v f α x +... (1) person

12 3D Vlasov-Poisson f t = v f + E vf Ω (0, T ], E = Φ Ω x (0, T ], Φ = IR 3 d3 v f 1 Ω x (0, T ]. Ω = Ω x IR 3

13 Vlasov Computational Methods Computational methods The VP system with the electrostatic force has been studied extensively for the simulation of collisionless plasmas. Numerical methods include but not limited Particle-In-Cell (PIC) (Birdsall, Langdon; Hockney, Eastwood, 1981) Semi-Lagrangian approach (Cheng and Knorr, 1976, Sonnendrücker, et al, F. Filbet, et al, Qiu and Christlieb ) Fourier-Fourier Spectral methods (Klimas et al), WENO FD with Fourier collocation (Zhou et al.), FEM, DG (see next page). For gravitational VP system, 1D problems, Fujiwara, 1981, White, 1986 Spherical stellar systems, Fujiwara, 1983 Stella disks, Nishida et al, Gravitational clustering, Bouchet, 1985 Computational challenges inlcude high dimensionality, large computational cost, conservation of physical quantities, etc.

14 DG reviewdiscontinuous Galerkin Method Invented by Reed and Hill (73) for neutron transport. Lesaint and Raviart (74). RKDG method by Cockburn and Shu (89, 90,...) for conservation laws. Elliptic and Parabolic problems, (IP methods), Babu ska et al. (73), Wheeler (78), Arnold (79), Bassi and Rebay (97), Cockburn and Shu (98), Arnold et al. (02)... DG methods for VP systems in electrostatic case have been considered Heath, Gamba, Morrison, Michler, JCP, Heath, 2007 Ayuso, Carrillo, Shu, KRM, to appear; preprint. Qiu, Shu, JCP, Rossmanith, Seal, JCP, 2011, Crouseilles et al., preprint.

15 DG Method Conservation Laws DG for nonlinear hyperbolic conservation laws For u t + f(u) = 0, the DG method is: to find u V(K), such that u t v da f(u) v da + f(u) n v ds = 0 K hold for any test function v V(K). K K K Upwinding

16 DG Method Advantages The main advantages of DG Real Boundary Conditions Use of FVM framework, convection-dominated problems. Flexibility with the mesh. (hanging nodes, nonconforming mesh) Compact scheme, highly parallelizable. Polynomials of different degrees in different elements, even non-polynomial basis. semi-discrete: M df dt = V (f), M 1 only once!

17 VP DG Error Estimates Error estimates for Vlasov Poisson (Ross Heath, PhD Thesis 2007): : the space of polynomials on a set K of degree less than or equal to r, and Non-Symmetric Interior Penalty (NIPG) method for the Poisson equation for and Broken Sobolev spaces H s (mesh) etc.

18 1D Vlasov-Poisson & Advection Equations Vlasov-Poisson: f t = vf x + Ef v Ω (0, T ] E = Φ x Ω x (0, T ], Φ xx = dv f 1 Ω x (0, T ] R Linear Vlasov-Poisson: Advection: (δf) t = v(δf) x + Ef 0 Ω x (0, T ] E = Φ x Ω x (0, T ] Φ xx = R dv δf Ω x (0, T ] (δf) t = v(δf) x Ω (0, T ] Ω x = [0, L], Ω = Ω x IR

19 ICs and BCs f(x, v, t) = f 0 (v) + δf(x, v, t) δf(x, v, 0) = A cos(kx) f 0 (v), δf(0, v, t) = δf(l, v, t), Φ(0, t) = Φ(L, t) = 0, Note, δf(l, v, t) need not be small. Sample equilibria: Maxwellian : f M = 1 2π e v2 /2 Lorentzian : f L = 1 π 1 v

20 Advection ρ tot (x, t) = 1 dv f(x, v, t) = 1 dv f(x vt, v) = A dv cos [k(x vt)] f 0(v),

21 Maxwellian Advection Choose: f 0 = f M, A = 0.1, k = 0.5, L = 4π ρ tot (x, t) = A cos (kx) e k2 t 2 /2 max x ρ tot (x, t) = 0.1 e t2 /8

22 Maxwellian Advection log 10 (max x ρ tot (x, t) ) vs. t exact solution (solid), (N hx, N hv ) = (500,400)(dot), (1000,800)(dash-dot), (2000,1600)(dash-dot-dot), (4000, 400) (short dash), (8000, 400) (long dash).

23 Lorentzian Advection Choose: f 0 = f L, A = 0.01, k = 1/8, 1/6, 1/4, 1/2, L = 16π, 12π, 8π, 4π T = 75, 75, 50, 50, (N x, N v ) = (1000, 2000) ρ tot (x, t) = A cos (k x ) e kt max x ρ tot (x, t) = 0.01 e kt

24 Lorentzian Advection log 10 (max x ρ tot (x, t) ) vs. t k=1/8 (left) and k=1/6 (right) exact solution (dash), numerical solutions (solid). k=1/4 (left) and k=1/2 (right); exact solution(dash), numerical solutions (solid).

25 Landau Damping Assume: f(x, v, t) = f 0 (v) + δf(x, v, t), δf(x, v, t) exp(ikx iωt) Plasma Dispersion Relation : ε(k, ω) = 1 1 k 2 k real and positive, ω in UHP f 0 (v) (v ω/k) dv, Stable and unstable eigenmodes (and embedded modes) if they exist satisfy ε(k, ω) = 0 ω(k) = ω R (k) + iγ(k) Landau damping comes from analytically continuing into LHP (deforming the contour). Not an eigenmode! Time asymptotics.

26 Landau Damping Maxwellian

27 Contour plots (left) and cross-sectional plots (right), x = 2π, for δf at t = 0, t = 25, t = 50, t = 75 (descending order).

28 Landau Damping Maxwellian Decay Rate Time decay plots of fundamental mode under mesh refinement: (N hx, N hv ) =(250, 200) (top left), (500, 400) (top right), (1000, 800) (bottom left) and (2000, 1600) (bottom right). The theoretical decay rate is to three decimal-digit accuracy.

29 Landau Damping with Lorentzian Plasma Dispersion Function: ε(k, ω) = πk 2 v (v 2 + 1) 2 (v u) dv, Residue calculus implies: ε(k, u) = 1 1 k 2 (u + i) 2. ε = 0 and u = ω/k implies ω = ω R + iγ = ±1 ik,

30 Landau Damping with Lorentzian: γ = k Decay plots of fundamental modes: k=1/8 (top left), k=1/6 (top right), k=1/4 (bottom left) and k=1/2 (bottom right).

31 Recurrence in Advection Given a map on a bounded domain D, f t : D D, with f measure preserving homeomorphism recurrence.

32 Recurrence time (Cheng and Knorr, 76) Cheng-Knorr Recurrence Time ρ(x, t) = j f(x, v j, t) v = j f 0 (x v j t, v j ) v = j Af eq (v j ) cos(k(x v j t)) v = j Af eq (v j ) cos(kx k(j + 1/2) vt) v This is a periodic function in time with period T R = 2π k v. In this section, we consider the standard RKDG methods for this equation with upwind numerical fluxes.

33 DG Recurrence Time Q 1 f h = f i 1 4,j+ 1 4 χ 1 (x, v) + f i 1 4,j 1 4 χ 2 (x, v) +f i+ 1 4,j+ 1 4 χ 3 (x, v) + f i+ 1 4,j 1 4 χ 4 (x, v), ( x xi χ 1 (x, v) = 4 1 ) ( v vj + 1 x i 4 v j 4 ( x xi χ 2 (x, v) = 4 1 ) ( v vj x i 4 ) 1 v j 4 ( x xi χ 3 (x, v) = ) ( v vj + 1 ) x i 4 v j 4 ( x xi χ 4 (x, v) = ) ( v vj 1 x i 4 v j 4 ) ) ) f ij = (f i 1/4,j+1/4, f i 1/4,j 1/4, f i+1/4,j+1/4, f i+1/4,j 1/4 ) T

34 DG Recurrence Time Q 1 df ij dt = v x ( Sm f ij + T m f i 1,j ) = v x ( Sm + T m e ik x) f ij S m = T m = m m m m m m m m m m m m 8, m m m 1, m 8 with m = 2j N v 1 = 1, 3, 5...

35 DG Recurrence Time Q 1 The initial condition is f ij (0) = Re(Ae ikx i Λ), where Λ = (e ik x/4 f eq (v j+1/4 ), e ik x/4 f eq (v j 1/4 ), e ik x/4 f eq (v j+1/4 ), e ik x/4 f eq (v j 1/4 )) T Hence the general expression for the numerical solution is f ij (t) = Re(e ikx i (a 1 e η 1t V 1 + a 2 e η 2t V 2 + a 3 e η 3t V 3 + a 4 e η 4t V 4 )) where η 1,... η 4 are eigenvalues of G j, and V 1,... V 4 are corresponding eigenvectors. Eigenvectors independent of m = 2j N v 1 Exact solution: Recurrence T R 2π/k v, modulation, and decay O(k 2 x 2 ).

36 DG Recurrence Time Q ρ ρ t t ρ 10-4 ρ t t Figure: Top left: Maxwellian, Q 1. Top right: Lorentzian, Q 1. Bottom left: Maxwellian, Q 2. Bottom right: Lorentzian, Q 2.

37 Landau Damping Q 2 Recurrence Time E_max t H L = 1 2 4π 0 dx IR dv v (δf)2 f π 4π 0 dx E2.

38 Nonlinear Computations Analysis of Results

39 Nonlinear Landau Damping Maxwellian, amplitude A =.5: k=.5 (top left), k=1 (top right), k=1.5 (bottom left) and k=2 (bottom right). Bounce time 40.

40 Nonlinear Landau Damping Maxwellian, amplitude A =.5. First mode. γ smaller than linear Landau damping because nonlinear coupling matters early.

41 Nonlinear Two-Stream Instability Equilibrium: f T S (v) = 1 2π v 2 e v2 /2 Manipulations: where z = ω/k. Plasma Z-function: ε = 1 2 k 2 [ 1 2z 2 + 2zZ(z) ( 1 z 2)]. Z(z) = 1 π e w2 dw w z = 2ie z2 iz e t2 dt first expression I(z) > 0 and the value of Z for I(z) < 0 is obtained by analytic continuation; second expression valid for all complex z good for numerics. ε = 0 implies instability! γ agrees!

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43 Invariants Particle Number: Total Momentum: Total Energy: L N = P = L 0 dx L 0 dx R R dv f(x, v, t), dv vf(x, v, t), H = 1 2 dx 0 R dv v 2 f(x, v, t) Casimir Invariants: C = L 0 dx R dv C(f). L 0 dx E(x, t) 2,

44 Invariants Relative Error Total particle number (left). Total momentum (right). Enstrophy (left). Total energy (right).

45 BGK Mode Potential The electrostatic potential up to Φ(x, t = 100)

46 Scatter Plot f versus E(x, v) At t = 100 for every x, v, know Φ E(x, v) = v 2 /2 Φ(x, 100). Make scatter plot of 9 million pairs (x, v) of f 100 versus E(x, v): f 100 a graph over E(x, v) to within line thickness. Green positive velocities; red negative velocities.

47 Scatter Plot Detail Blow-up of f 100 (E) near E = 0. Is cusp universal trapping feature?

48 BGK Modeling Model Distribution: f fit = A(E + Φ M )(E + E )e βe. Here Φ M = max(φ). Since E = v 2 /2 Φ, min(e) = Φ M. f > 0 E > Φ M Rough guess: Φ M = 1 and E = 2 uniformly good fit. f (E M ) = 0. For β = 1, E M = 1/γ, where γ is the golden mean!

49 Pseudo-potential ρ(φ) = Poisson s Equation: IR dvf(e) = Φ Φ xx = ρ(φ) = dv dφ de f 0 (E) 2(E + Φ) Integrable Newton s second law: Φ x, x t. Oscillation if pseudo-potential V has local minimum etc. Compares well.

50 Phys. Plasmas 16, Dynamically Accessible IC Vlasov with Drive: f t = vf x + ( E + E d (x, t) ) f v, External Drive: E x = 1 IR dv f E d (x, t) = A d (t) cos(kx ωt) Drive Created IC: A d (t) =.052 and T d = 200 Johnston et al., Afeyan, Rose, PJM,...

51 Weak Drive: E(t) = E(t + T ) A d (t) =.052 and T d = 200 Appears to settle into periodic orbit travelling BGK hole.

52 Strong Drive E(t) at x center point Basis functions P E(t) t A d (t) =.4 and T d = 200 Higher Order Periodic/Quasiperiodic Orbit: E(t) = A(t)E 0 (t) A(t) = A(t + T/4) with E 0 (t) = E 0 (t + T ) E 0 (t) like weak drive

53 Strong Drive Fourier

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55 Open Mathematics Problems Prove nonlinear Landau damping rate, growth, bounce say anything about general phenomenology. Prove stability of any BGK mode. Mine? Prove weak asymptotic stability. Prove existence/nonexistence of cusp. Prove existence of weak drive periodic orbit. Stability. Weak asymptotic stability. Prove existence of strong drive periodic/quasiperiodic orbit. Stability. Weak asymptotic stability.

56 How? Finite-Dimensional Hamiltonian Systems: periodic orbits near equilibria Lyapunov, Weinstein, Moser,... variational methods Rabinowitz, Ekland,... Infinite-Dimensional Hamiltonian VP-Like Systems: Hamilton-Jacobi Variational Principle for VP PJM,... tutorial web page, online ICERM lecture techniques: viscosity solutions, weak KAM,... Villiani, Gangbo, Li,... Time is Ripe!