Semi-Lagrangian Formulations for Linear Advection and Applications to Kinetic Department of Mathematical and Computer Science Colorado School of Mines joint work w/ Chi-Wang Shu Supported by NSF and AFOSR. CSCAMM, University of Maryland College Park 1 / 39
Outline Background: numerical s for equations Semi-Lagragian finite difference s for linear advection equations Ongoing and future work / 39
VP system The Vlasov-Poisson (VP) system, f t + p xf + E(t, x) p f = 0, (1) E(t, x) = x φ(t, x), x φ(t, x) = ρ(t, x). () f (t, x, p): the probability of finding a particle with velocity p at position x at time t. ρ(t, x) = f (t, x, p)dp - 1: charge density 3 / 39
Numerical approach: Lagrangian vs. Eulerian vs. semi-lagrangian Lagrangian: tracking a finite number of macro-particles. e.g., PIC (Particle In Cell) dx dt = v, dv dt = E (3) Eulerian: fixed numerical mesh e.g., finite difference WENO, finite volume, finite element, spectral. Semi-Lagrangian: e.g., finite difference, finite volume, finite element, spectral. 4 / 39
Strang splitting for solving the Vlasov equation f t + v xf + E(t, x) v f = 0, Nonlinear Vlasov eqation a sequence of linear equations. 1-D in x and 1-D in v: f t + v f x = 0, (4) f t + E(t, x) f v = 0. (5) 5 / 39
SL schemes 1 Solution space: point values, or cell averages, or piecewise polynomials living on fixed Eulerian grid. Evolution: tracking characteristics backward in time. 3 Project the evolved solution back onto the solution space. Remark: The only error in time comes from tracking characteristics backward in time. The scheme is not subject to CFL time step restriction. 6 / 39
Various formulations of difference schemes difference (point values) Scheme I: advective scheme the solution is evolved along characteristics (in Lagrangian spirit) approximating advective form of equation f t + af x = 0. Scheme II: convective scheme the solution is evolved over fixed point (in Eulerian spirit) approximating conservative form of equation f t + (af ) x = 0. a being a constant, with possible extension to a = a(x, t) (relativistic Vlasov equation). 7 / 39
Scheme I: advective scheme f n+1 i = f (x i, t n+1 ) = f (x i, t n ) = f (x i ξ 0 x, t n ), ξ 0 = a t x When ξ 0 [0, 1 ] t n+1 t n x 0 x i x i 1 x i x i+1 x i+ x N ξ 0 [0, 1 ] 8 / 39
Third order example f n+1 i = fi n + ( 1 6 f i n + fi 1 n 1 f i n 1 3 f i+1)ξ n 0 + ( 1 f i 1 n fi n + 1 f i+1)ξ n 0 + ( 1 6 f i n 1 f i 1 n + 1 f i n 1 6 f i+1)ξ n 0, 3 (6) Method 1: apply WENO interpolation on (6). No mass conservation. 9 / 39
Matrix vector form f n+1 i with matrix A L 3 = = = f n i + ξ 0 (f n i, f n i 1, f n i, f n i+1) A L 3 (1, ξ 0, ξ 0), (7) 1/6 0 1/6 1 1/ 1/ 1/ 1 1/ 1/3 1/ 1/6 1/6 0 1/6 5/6 1/ 1/3 1/3 1/ 1/6 0 0 0 0 0 0 1/6 0 1/6 5/6 1/ 1/3 1/3 1/ 1/6. 10 / 39
Conservative formulation Rewrite the advective scheme into a conservative form with f n+1 i = fi n ξ 0 ((fi 1, n fi n, fi+1) n C3 L (1, ξ 0, ξ0) (fi, n fi 1, n fi n ) C3 L (1, ξ 0, ξ0) ) = f n i ξ 0 (ˆf n i+1/ ˆf n i 1/ ) 1 1 C3 L 6 0 6 = 5 1 6 1 3 1 1 3 1 6 Method : apply WENO reconstructions on flux functions ˆf n i+1/ The scheme is locally conservative. However, such splitting can NOT be generalized to equations with variable coefficients.. 11 / 39
Scheme II: conservative scheme Integrating the conservative form of PDE over [t n, t n+1 ] at x i gives f n+1 i = fi n (. = fi n f t + (af ) x = 0, t n+1 t n F x x=xi? = fi n 1 x ( ˆF i+ 1 af (x, τ)dτ) x x=xi ˆF i 1 ) + O( x k ) where F(x) = t n+1 t n af (x, τ)dτ, and ˆF i± 1 are numerical fluxes. 1 / 39
F(x) t n+1 F(x i ) = af (x i, τ)dτ = t n xi x i f (ξ, t n )dξ, by applying the Divergence Theorem on Ω (f t + (af ) x )dx = 0. t n+1 t n x 0 x i x i 1 x i x i+1 x i+ x N 13 / 39
F(x) F(x i ) = t n+1 t n af (x i, τ)dτ = xi x i f (ξ, t n )dξ, by applying the Divergence Theorem on Ω (f t + (af ) x )dx = 0. t n+1 Ω t n x 0 x i x i 1 x i x i+1 x i+ x N xi 13 / 39
ˆF i± 1 Let H(x) to be a function s.t. then F(x) = 1 x+ x H(ξ)dξ x x x F x x=xi = 1 x (H(x i+ 1 ) H(x i 1 )). Let numerical fluxes ˆF i± 1 = H(x i± 1 ). 14 / 39
f n+1 i = fi n (. = fi n F x t n+1 t n A first order scheme af (x, τ)dτ) x = fi n 1 x (F(x i) F(x i 1 )), if a > 0 = fi n 1 xi xi 1 x ( f (ξ, t n )dξ f (ξ, t n )dξ) x i x i 1 Remark. When a is a constant and cfl < 1, the scheme reduces to f n+1 i = f n i a t x (f i n fi 1) n which is the first order upwind scheme. 15 / 39
High order WENO reconstructions High order WENO reconstructions are applied in the following reconstructions { } N xi { } {f (x i, t n )} N WENO i=1 f (ξ, t n WENO N )dξ ˆF i+ 1 x i i=1 Computational expensive: two weno reconstruction procedures The reconstruction stencil is widely spread (not compact), leading to instability of algorithm. i=1 16 / 39
{f (x i, t n )} N i=1 {f (x i, t n )} N i=1 WENO { xi f (ξ, t n )dξ x i WENO/ENO } N i=1 WENO { ˆF i+ 1 { ˆF i+ 1 } N } N i=1 i=1 17 / 39
Third order WENO reconstruction when a = 1, dt < dx. Consider two substencils S 1 = {x i 1, x i }, S = {x i, x i+1 }. Let ξ = dt dx, the third order reconstruction from the three point stencil {x i 1, x i, x i+1 } ˆF i+ 1 = γ 1 ˆF (1) + γ i+ 1 ˆF () i+ 1 where the linear weights γ 1 = 1 ξ 3, γ = + ξ 3, 18 / 39
and ˆF (1) i+ 1 ˆF () i+ 1 = ( ( 3 ξ + 1 ξ )f n i = (( 1 ξ + 1 ξ )f n i + ( 1 ξ + 1 ξ )f n i 1)dx ( 1 ξ + 1 ξ )f n i+1)dx Idea of WENO: adjust the linear weighting γ i to a nonlinear weighting w i, such that w i is very close to γ i, in the region of smooth structures, w i weights little on a non-smooth sub-stencil. 19 / 39
Smoothness indicator: Nonlinear weights β 1 = (f n i 1 f n i ), β = (f n i f n i+1) w 1 = γ 1 /(ɛ + β 1 ), w = γ /(ɛ + β ) Normalized nonlinear weights w i : w 1 = w 1 /( w 1 + w ), w = w /( w 1 + w ) 0 / 39
Method 3 of scheme II 1 Reconstruction numerical fluxes ˆF i± 1 reconstruction of {fi n } N i=1 f n+1 i = fi n 1 x ( ˆF i+ 1 from WENO/ENO ˆF i 1 ), Compare with the of line approach (Runge Kutta time discretization), the time integration of the PDE here is exact. The scheme can be extended to accommodate extra large time step evolution and to the case of variable coefficient a(x, t). 1 / 39
Method 1 & : Approximates the advective form of PDE Frame of reference: Lagrangian Method is the conservative formulation of Method 1 for equations with constant coefficients Method 1 can be generalized to equations with variable coefficients Method 3: Approximates the conservative form of PDE Frame of reference: Eulerian conservative scheme extends to equations with variable coefficients / 39
Numerical s Method 1,, 3 with fifth order WENO reconstruction Linear advection equation Rigid body rotation Vlasov Poisson system 3 / 39
Linear transport: u t + u x + u y = 0 Table: Order of accuracy with u(x, y, t = 0) = sin(x + y) at T = 0. dt = 1.1dx = 1.1dy. 1 3 mesh error order error order error order 0 0 6.03E-4 8.8E-4 7.94E-4 40 40.4E-5 4.75.6E-5 4.97.51E-5 4.98 60 60 3.10E-6 4.88 3.44E-6 5.00 3.9E-6 5.01 80 80 7.50E-7 4.93 8.16E-7 5.00 7.80E-7 5.00 4 / 39
Rigid body rotation: u t yu x + xu y = 0 Figure: Initial profile and numerical solution of 1 from the numerical mesh 100 100, dt = 1.1dx = 1.1dy at T = 1π 5 / 39
Rigid body rotation: u t yu x + xu y = 0 Figure: Method and 3 with the numerical mesh 100 100, dt = 1.1dx = 1.1dy at T = 1π. 6 / 39
Rigid body rotation: u t yu x + xu y = 0 Figure: Method 1 (left) and 3 (right). 7 / 39
Rigid body rotation: u t yu x + xu y = 0 Figure: Method 1 (left) and 3 (right). 8 / 39
Landau damping Consider the VP system with initial condition, f (x, v, t = 0) = 1 (1 + αcos(kx))exp( v ), (8) π 9 / 39
Weak Landau damping: α = 0.01, k = Figure: Time evolution of L norm of electric field. 30 / 39
Weak Landau damping (cont.) Figure: Time evolution of L 1 (upper left) and L (upper right) norm, discrete energy (lower left), entropy (lower right). 31 / 39
Strong Landau damping: α = 0.5, k = Figure: Time evolution of the L norm of the electric field. 3 / 39
Strong Landau damping (cont.) Figure: Time evolution of L 1 (upper left) and L (upper right) norm, discrete energy (lower left), entropy (lower right). 33 / 39
Two-stream instability Consider the symmetric two stream instability, the VP system with initial condition f (x, v, t = 0) = 7 π (1 + 5v )(1 + α((cos(kx) +cos(3kx))/1. + cos(kx))exp( v ), with α = 0.01, k = 0.5 and the length of domain in x direction L = π k. 34 / 39
Figure: Two stream instability T = 53. The numerical solution of 1 (left) and Method 3 (right) with the numerical mesh 64 18. 35 / 39
Figure: Third order semi-lagrangian WENO. Two stream instability T = 53. The numerical mesh is 64 18 (left) and 56 51 (right). 36 / 39
Figure: Time development of L 1 (upper left) and L (upper right) norm, discrete energy (lower left), entropy (lower right). 37 / 39
Ongoing and future work Algorithm: of variable coefficients Truely multi-dimensional formulation of semi-lagrangian scheme evolving point values. Application advection in incompressible flow relativistic Vlasov equations 38 / 39
THANK YOU! 39 / 39