Alexander Ostermann University of Innsbruck, Austria Joint work with Lukas Einkemmer Verona, April/May 2017
Plasma the fourth state of matter 99% of the visible matter in the universe is made of plasma Source: Hubble telescope Source: Luc Viatour / www.lucnix.be
Vlasov equation Evolution of (collisionless) plasma is described by t f (t, x, v) + v x f (t, x, v) + F v f (t, x, v) = 0, where f... particle density function v... velocity x... position F... force field Coupled to the self-consistent electric (and magnetic) field via F = e (E + v B) m yields the Vlasov Poisson (Vlasov Maxwell) equations. Applications: Tokamaks (fusion), plasma-laser interactions,...
Vlasov equation, cont. Anatoly Alexandrovich Vlasov (1908 1975), Moscow State U The vibrational properties of an electron gas (1938, 1968) e t f (t, x, v )+v x f (t, x, v )+ E +v B v f (t, x, v ) = 0 m
Filamentation of phase space Simulation of the filamentation of phase space as is studied, for example, in the context of Landau damping. The horizontal axis represents space and the vertical one velocity.
Bump-on-tail instability The bump-on-tail instability leads to a traveling vortex in phase space. The horizontal axis represents space and the vertical one velocity.
Outline Strang splitting for Vlasov-type equations dg space discretization Numerical examples References
Abstract formulation Abstract initial value problem where t f (t) = (A + B)f (t), f (0) = f 0, A is a linear differential operator; the nonlinear operator B has the form Bf = B(f )f, where B(f ) is an (unbounded) linear operator. Formulation comprises the Vlasov Poisson and the Vlasov Maxwell equations as special cases: take A = v x and choose B appropriately
Example: Vlasov Poisson equation in 1+1 d Recall the abstract formulation t f (t) = (A + B)f (t), f (0) = f 0. Example: Vlasov Poisson equations in 1+1 dimensions t f (t, x, v) = v x f (t, x, v) E(f (t,, ), x) v f (t, x, v) x E(f (t,, ), x) = f (t, x, v) dv 1, E = x Φ R f (0, x, v) = f 0 (x, v) with periodic boundary conditions in space f (t, x, v) = f (t, x + L, v).
Strang splitting Problem: t f (t) = (A + B)f (t), Bf = B(f )f. Denote f k f (t k ) at time t k = kτ with step size τ. Solve t f = Af, t g = B k+1/2 g, where B k+1/2 B ( f (t k + τ 2 )) and let f k+1 = S k f k, where the (nonlinear) splitting operator S k is given by S k = e τ 2 A e τb k+1/2 e τ 2 A. For grid-based Vlasov solvers: Cheng and Knorr 1976; Mangeney et al. 2002
Computational efficiency and stability Splitting is very efficient in our application: t f = v x f f (τ, x, v) = e τv x f (0, x, v) = f (0, x τv, v) and t g = E k+1/2 (x) v g g(τ, x, v) = e τe k+1/2(x) v g(0, x, v) = g(0, x, v τe k+1/2 (x)) imply S k = 1. Moreover, we have ( ) e τb k+1/2 e τb(f (t k+τ/2)) e τa f (t k ) Cτ f k+1/2 f (t k +τ/2) (requires some smoothness of f ).
Consistency technical tools Lemma [Gröbner Alekseev formula] Consider Then f (t) = G(f (t)) + R(f (t)), f (0) = f 0 g (t) = G(g(t)) with solution E G (t, g 0 ) f (t) = E G (t, f 0 ) + t 0 2 E G (t s, f (s)) R (f (s)) ds. Proof. Fundamental theorem of calculus + trick. Lemma. Let E generate a (semi)group. Then e τe = m 1 k=0 τ k k! E k + τ m E m 1 e (1 s)τe 0 s m 1 (m 1)! ds Proof. Integration by parts.
Proof of the Gröbner Alekseev formula Fix t and let u(t) be a solution of u (t) = G (u(t)). Next differentiate the identity with respect to s to get E G (t s, u(s)) = u(t) 1 E G (t s, u(s)) + 2 E G (t s, u(s)) G (u(s)) = 0. The initial value of u is now chosen such that u(s) = f (s) which implies 1 E G (t s, f (s)) + 2 E G (t s, f (s)) G (f (s)) = 0.
Proof of the Gröbner Alekseev formula, cont. Let ϕ(s) = E G (t s, f (s)). By the fundamental theorem of calculus f (t) E G (t, f 0 ) = = = t 0 t 0 t 0 ϕ (s) ds ( ) 1 E G (t s, f (s)) + 2 E G (t s, f (s))f (s) ds 2 E G (t s, f (s)) R (f (s)) ds, where we have used f (s) = G(f (s)) + R(f (s)) and 1 E G (t s, f (s)) + 2 E G (t s, f (s)) G (f (s)) = 0.
Consistency, cont. Expansion of the exact solution f (τ) = E B (τ, f 0 ) + + + τ τ1 0 0 τ τ1 τ2 0 0 τ 0 2 E B (τ τ 1, f (τ 1 ))AE B (τ 1, f 0 ) dτ 1 2 E B (τ τ 1, f (τ 1 ))A 2 E B (τ 1 τ 2, f (τ 2 ))AE B (τ 2, f 0 )dτ 2 dτ 1 ( 2 ) 2 E B (τ k τ k+1, f (τ k+1 ))A f (τ 3 ) dτ 3 dτ 2 dτ 1, 0 where τ 0 = τ. k=0 Expansion of the numerical solution S k f 0 = e τ 2 A e τb k+1/2 e τ 2 A f 0 S 0 f 0 = e τb 1/2 f 0 + τ { } A, e τb 1/2 f0 + τ 2 { { }} A, A, e τb 1/2 f0 +R 3 f 0 2 8
Consistency, cont. Compare the terms by employing quadrature rules (Jahnke, Lubich 2000). Treat τ term with trapezoidal rule τ 0 2 E B (τ τ 1, f (τ 1 ))AE B (τ 1, f 0 ) dτ 1 = and compare with τ 2 2E B (τ, f 0 )Af 0 + τ 2 AE B(τ, f 0 )f 0 + R 1 τ { } A, e τb 1/2 f0 = τ 2 2 A e τb 1/2 f 0 + τ 2 e τb 1/2 Af 0
Consistency, cont. Estimate the difference ( e τb 1/2 2 E B (τ, f 0 ) ) Af 0 = 2 ( e τb 1/2 E B (τ, f 0 ) ) Af 0. Apply again the variation of constants formula E B (τ, f 0 ) e τb 1/2 f 0 = τ 0 e (τ s)b 1/2 (B B 1/2 )E B (s, f 0 )ds. Gives condition on B 1/2. Treat all terms in this way and estimate remainders.
Regularity of exact solution Consider Vlasov Poisson problem as before. Theorem (Glassey 1996) Assume that f 0 C m per,c is non-negative, then f C m (0, T ; C m per,c), E(f (t,, ), x) C m (0, T ; C m per). Moreover, there exists Q(T ) such that for all t [0, T ] and x R it holds that supp f (t, x, ) [ Q(T ), Q(T )]. Important result: Regularity of initial data is preserved.
Abstract convergence result Theorem (L. Einkemmer, A.O., SIAM J. Numer. Anal., 2014a) Let f 0 C 3 per,c be non-negative f k+1/2 f (t k + τ ) Cτ 2 Then Strang splitting for the Vlasov Poisson equations is second-order convergent. Possible choices for f k+1/2 : first-order Lie Trotter splitting f k+1/2 = e τ 2 B(fk) e τ 2 A f k due to the special structure of the E field f k+1/2 = e τ 2 A f k linear extrapolation using f k 1 and f k
Outline Strang splitting for Vlasov-type equations dg space discretization Numerical examples References
dg approximation in space and velocity Choose grid in (x, v) plane with cells R ij. Project onto space of Legendre polynomials of order l. The projection is given by Pg Rij = l l b ij km P(1) k k=0 m=0 (x)p(2) m (v) with coefficients b ij (2k + 1)(2m + 1) km = h x h v P (1) k (x)p(2) m (v)g(x, v) d(x, v), R ij where P (1) k and P m (2) are translated and scaled Legendre polynomials.
Space discretization Pe τ 2 A g(x, v) = Pg ( x τ v, v) 2 cell 1 cell 2 cell 1 cell 2 Polynomial approximation of functions with a small jump discontinuity; Jump heights ε (k) = g (k) (x 0 +) g (k) (x 0 ) satisfy ε (k) ch l k+1 for all k {0,..., l}; Then g (k) (Pg) (k) Ch l k+1, 0 k l.
Space and velocity discretization Pe τ 2 A g(x, v) = Pg ( x τ v, v) 2 Pe τb(pf k+1/2) g(x, v) = Pg ( x, v τe k+1/2 (x) ) evaluate integrals exactly with Gauss Legendre quadrature Spatially discretized Strang splitting: S k = Pe τ 2 A Pe τb(pf k+1/2) Pe τ 2 A
Convergence of full discretization Theorem (L. Einkemmer, A.O., SIAM J. Numer. Anal., 2014b) Suppose that f 0 C max{l+1,3}, nonnegative and compactly supported. Then ( n 1 ) ) S k Pf 0 f (nτ) (τ C 2 + hl+1 + h l+1. τ k=0 Proof. Combine time and space error
Stability, revisited In advection dominated problems instabilities can occur if the numerical integration is not performed exactly. Example (Molenkamp Crowley test problem) t f (t, x, y) = 2π(y x x y )f (t, x, y) f (0, x, v) = f 0 (x, y), with off-centered Gaussian as initial value. Initial value turns around origin with period 1. Finite element schemes turn out to be unstable for most quadrature rules [Morton, Priestley, Süli (1988)].
Molenkamp Crowley test problem, cont. Morton, Priestley, Süli: Stability of the Lagrange Galerkin method with non-exact integration, RAIRO (1988)
Molenkamp Crowley test problem, cont. 1 1 max: 1.0 min: -0.002 max: 0.998 min: -0.01 0.5 0.5 0.9 0.7 0.1 y 0 0.9 0.7 0.1 y 0 0.4 0.4 0-0.5-0.5 0-1 -1-0.5 0 0.5 1 x -1-1 -0.5 0 0.5 1 x Projected initial value (left); numerical solution after 60 revolutions with τ = 0.02, N x = N v = 40, and l = 2 (right). Our dg method is stable.
Outline Strang splitting for Vlasov-type equations dg space discretization Numerical examples References
Example: nonlinear Landau damping Vlasov Poisson equations in 1+1 dimensions on [0, 4π] R together with the initial value f 0 (x, v) = 1 ( e v 2 /2 1 + α cos x ). 2π 2 Landau damping is a popular test problem. weak or linear Landau damping α = 0.01 strong or nonlinear Landau damping α = 0.5 Consider Strang and Lie Trotter splitting S L,k = e τb k e τa.
Nonlinear Landau damping; time error error (discrete L1 norm) 1 0.1 0.01 0.001 0.0001 1e-005 Strang splitting Lie-Trotter splitting order 1 order 2 1e-006 0.015625 0.03125 0.0625 0.125 0.25 0.5 1 step size Error (in discrete L 1 norm) of the particle density function f (1,, ) for Strang and Lie Trotter splitting, respectively. h
Nonlinear Landau damping; space error error (infinity norm) 10 1 0.1 0.01 0.001 0.0001 1e-05 1e-06 l=0 l=1 l=2 order 1 order 2 order 3 1e-07 32 64 128 Spatial error (in maximum norm) of the particle density function for dg discretizations of Strang splitting. N
Recurrence phenomenon Most easily understood for a transport equation. Consider t f = v x f, f 0 (x, v) = e v 2 /2 2π ( 1 + 0.01 cos(0.5x) ) with exact solution f (t, x, v) = e v 2 /2 2π ( 1 + 0.01 cos(0.5x 0.5vt) ). The electric energy E(t) = 4π decays exponentially in time. 0 E(t, x) 2 dx = π 1250 e 0.25t2
Recurrence, cont. For the numerical approximation of f (t, x, v) = e v 2 /2 2π ( 1 + 0.01 cos(0.5x 0.5vt) ). consider a piecewise constant approximation in v with grid size h v. Let the time t p be given by h v t p = 4π, v j t p = jh v t p = 4jπ. Obviously, the numerical solution is periodic with period t p. The numerical approximation to the electric energy E is thus periodic in time as well.
Recurrence for the advection equation 1 0.01 Nv=32, l=0 Nv=64, l=0 Nv=32, l=2 exact solution electric energy 0.0001 1e-06 1e-08 1e-10 1e-12 1e-14 1e-16 0 10 20 30 40 50 60 70 80 90 100 t There is no periodicity in the approximation of degree 2. Nevertheless, a recurrence-like effect is still visible.
Recurrence, weak Landau damping The electric energy decays asymptotically as e 2γt, γ 0.1533. We compare our numerical solutions with this function. 1 0.01 0.0001 Nv=64, l=1 Nv=64, l=2 Nv=64, l=4 analytic decay rate 1 0.01 0.0001 Nv=128, l=1 Nv=128, l=2 analytic decay rate electric energy 1e-06 1e-08 1e-10 electric energy 1e-06 1e-08 1e-10 1e-12 1e-12 1e-14 1e-14 1e-16 0 10 20 30 40 50 60 70 80 90 100 1e-16 0 10 20 30 40 50 60 70 80 90 100 t t The decay of the electric energy is shown for N x = N v = 64 (left) and N x = N v = 128 (right). In all cases a relatively large time step of τ = 0.2 is employed.
A strategy to suppress the recurrence effect Recurrence is not a consequence of the lack of accuracy of the underlying space discretization, but a result of aliasing. higher and higher frequencies in phase space are created; aliasing of the high frequencies introduces an error in the macroscopic quantities (such as the electric energy). Strategy to avoid recurrence (Einkemmer, A.O., EJPD 2014): damp highest frequencies; in our dg implementation: FFTs of coefficients of Legendre polynomials; after a cutoff time t c, cut off the highest M frequencies cutoff (t c,m)
Recurrence, weak Landau damping piecewise constant approximation, N=64 electric energy 0.01 0.0001 1e-06 1e-08 1e-10 1e-12 1e-14 1e-16 0 50 100 150 200 time no cutoff, N=128 cutoff (1.2,5) piecewise linear aproximation, N=64 electric energy 0.01 0.0001 1e-06 1e-08 1e-10 1e-12 1e-14 1e-16 0 50 100 150 200 time no cutoff cutoff (1.2,5) piecewise quadratic Integration approximation, of Vlasov-type N=64 equations
Example: plasma echo phenomenon Initial value f 0 (x, v) = 1 2π e v 2 /2 (1 + α cos(k 1 x)) corresponds to an excitation with wavenumber k 1 at time t = 0. Furthermore, at time t = t 2 we excite a second perturbation with wavenumber k 2 ; that is, we superimpose α 2π e v 2 /2 cos(k 2 x) on the (numerical) solution at time t 2. Data: α = 10 3, k 1 = 12π/100, k 2 = 25π/100, and t 2 = 200
Echo, low resolution no cutoff, 5000 points in v-space (piecewise-constant) electric energy 0.0001 1e-06 1e-08 1e-10 1e-12 1e-14 echo 0 100 200 300 400 500 600 700 800 900 1000 time no cutoff, 500 points in v-space (piecewise-constant) electric energy 0.0001 1e-06 1e-08 1e-10 1e-12 1e-14 echo 0 100 200 300 400 500 600 700 800 900 1000 time electric energy 0.0001 1e-06 1e-08 1e-10 1e-12 1e-14 cutoff (1,20), 500 points in v-space (piecewise-constant) echo 0 100 200 300 400 500 600 700 800 900 1000 time
electric e 1e-08 1e-10 1e-12 Echo, higher 1e-14resolution 0 100 200 300 400 500 600 700 800 900 1000 time electric energy electric energy cutoff no cutoff, (1,20), 5000 points in in v-space (piecewise-constant) 0.0001 1e-06 echo 1e-08 1e-10 1e-12 1e-14 0 100 200 300 400 500 600 700 800 900 1000 time no cutoff, 1000 500 points in v-space (piecewise-constant) 0.0001 1e-06 echo 1e-08 1e-10 1e-12 1e-14 0 100 200 300 400 500 600 700 800 900 1000 time electric energy d) cutoff (1,20), 1000 500 points in v-space (piecewise-constant) 0.0001 1e-06 echo 1e-08 1e-10 1e-12 1e-14 0 100 200 300 400 500 600 700 800 900 1000 time
References L. Einkemmer, A. Ostermann. Convergence analysis of Strang splitting for Vlasov-type equations. SIAM J. Numer. Anal. 52, 140 155 (2014) https://arxiv.org/abs/1207.2090 L. Einkemmer, A. Ostermann. Convergence analysis of a discontinuous Galerkin / Strang splitting approximation for the Vlasov Poisson equations. SIAM J. Numer. Anal. 52, 757 778 (2014) https://arxiv.org/abs/1211.2353 L. Einkemmer, A. Ostermann. A strategy to suppress recurrence in grid-based Vlasov solvers. Eur. Phys. J. D 68, 197 (2014) https://arxiv.org/abs/1401.4809