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Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: theoretical analysis and application to the Vlasov-Poisson system. (English) Zbl 1273.65147
Summary: Semi-Lagrangian (SL) methods have been very popular in the Vlasov simulation community. In this paper, we propose a new Strang split SL discontinuous Galerkin (DG) method for solving the Vlasov equation. Specifically, we apply the Strang splitting for the Vlasov equation, as a way to decouple the nonlinear Vlasov system into a sequence of 1-D advection equations, each of which has an advection velocity that only depends on coordinates that are transverse to the direction of propagation. To evolve the decoupled linear equations, we propose to couple the SL framework with the semi-discrete DG formulation. The proposed SL DG method is free of time step restriction compared with the Runge-Kutta DG method, which is known to suffer from numerical time step limitation with relatively small CFL numbers according to linear stability analysis. We apply the recently developed positivity preserving (PP) limiter, which is a low-cost black box procedure, to our scheme to ensure the positivity of the unknown probability density function without affecting the high-order accuracy of the base SL DG scheme. We analyze the stability and accuracy properties of the SL DG scheme by establishing its connection with the direct and weak formulations of the characteristics/Lagrangian Galerkin method. The quality of the proposed method is demonstrated via basic test problems, such as linear advection and rigid body rotation, and via classical plasma problems, such as Landau damping and the two stream instability.

MSC:
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35Q83 Vlasov equations
65M25 Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
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[1] Arnold, D.; Brezzi, F.; Cockburn, B.; Marini, L., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM journal on numerical analysis, 39, 1749-1779, (2002) · Zbl 1008.65080
[2] Banks, J.; Hittinger, J., A new class of nonlinear finite-volume methods for Vlasov simulation, IEEE transactions on plasma science, 38, 2198-2207, (2010)
[3] Begue, M.; Ghizzo, A.; Bertrand, P.; Sonnendrucker, E.; Coulaud, O., Two-dimensional semi-Lagrangian Vlasov simulations of laser – plasma interaction in the relativistic regime, Journal of plasma physics, 62, 367-388, (1999)
[4] Besse, N.; Sonnendrucker, E., Semi-Lagrangian schemes for the Vlasov equation on an unstructured mesh of phase space, Journal of computational physics, 191, 341-376, (2003) · Zbl 1030.82011
[5] Carrillo, J.A.; Vecil, F., Nonoscillatory interpolation methods applied to Vlasov-based models, SIAM journal on scientific computing, 29, 1179-1206, (2007) · Zbl 1151.35397
[6] Cheng, C.; Knorr, G., The integration of the Vlasov equation in configuration space, Journal of computational physics, 22, 330-351, (1976)
[7] Cockburn, B.; Johnson, C.; Shu, C.-W.; Tadmor, E., Advanced numerical approximation of nonlinear hyperbolic equations, (1998), Springer New York · Zbl 0904.00047
[8] Cockburn, B.; Lin, S.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, Journal of computational physics, 84, 90-113, (1989) · Zbl 0677.65093
[9] Cockburn, B.; Shu, C.-W., TVB runge – kutta local projection discontinuous Galerkin finite element method for conservation laws II: general framework, Mathematics of computation, 411-435, (1989) · Zbl 0662.65083
[10] Cockburn, B.; Shu, C.-W., The runge – kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, Journal of computational physics, 141, 199-224, (1998) · Zbl 0920.65059
[11] Cockburn, B.; Shu, C.-W., Runge – kutta discontinuous Galerkin methods for convection-dominated problems, Journal of scientific computing, 16, 173-261, (2001) · Zbl 1065.76135
[12] Crouseilles, N.; Mehrenberger, M.; Sonnendrucker, E., Conservative semi-Lagrangian schemes for Vlasov equations, Journal of computational physics, 229, 1927-1953, (2010) · Zbl 1303.76103
[13] Douglas, J.; Russell, T., Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM journal on numerical analysis, 19, 871-885, (1982) · Zbl 0492.65051
[14] Filbet, F.; Sonnendrücker, E., Comparison of Eulerian Vlasov solvers, Computer physics communications, 150, 247-266, (2003) · Zbl 1196.82108
[15] Filbet, F.; Sonnendrücker, E.; Bertrand, P., Conservative numerical schemes for the Vlasov equation, Journal of computational physics, 172, 166-187, (2001) · Zbl 0998.65138
[16] Gottlieb, S.; Ketcheson, D.; Shu, C.-W., High order strong stability preserving time discretizations, Journal of scientific computing, 38, 251-289, (2009) · Zbl 1203.65135
[17] Heath, R.E.; Gamba, I.; Morrison, P.; Michler, C., A discontinuous Galerkin method for the vlasov – poisson system, Journal of computational physics, (2011) · Zbl 1244.82081
[18] Huot, F.; Ghizzo, A.; Bertrand, P.; Sonnendrucker, E.; Coulaud, O., Instability of the time splitting scheme for the one-dimensional and relativistic vlasov – maxwell system, Journal of computational physics, 185, 512-531, (2003) · Zbl 1073.82039
[19] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, Journal of computational physics, 126, 202-228, (1996) · Zbl 0877.65065
[20] Kubatko, E.; Dawson, C.; Westerink, J., Time step restrictions for runge – kutta discontinuous Galerkin methods on triangular grids, Journal of computational physics, 227, 9697-9710, (2008) · Zbl 1154.65071
[21] LeVeque, R., High-resolution conservative algorithms for advection in incompressible flow, SIAM journal on numerical analysis, 627-665, (1996) · Zbl 0852.76057
[22] Lin, S.; Rood, R., Multidimensional flux-form semi-Lagrangian transport schemes, Monthly weather review, 124, 2046-2070, (1996)
[23] Morton, K.; Priestley, A.; Suli, E., Stability of the lagrange – galerkin method with non-exact integration, Modélisation mathématique et analyse numérique, 22, 625-653, (1988) · Zbl 0661.65114
[24] Qiu, J.-M.; Christlieb, A., A conservative high order semi-Lagrangian WENO method for the Vlasov equation, Journal of computational physics, 229, 1130-1149, (2010) · Zbl 1188.82069
[25] Qiu, J.-M.; Shu, C.-W., Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow, Journal of computational physics, 230, 863-889, (2011) · Zbl 1391.76489
[26] J.-M. Qiu, C.-W. Shu, Conservative semi-Lagrangian finite difference WENO formulations with applications to the Vlasov equation, Communications in Computational Physics (2011), submitted for publication. · Zbl 1388.65066
[27] Restelli, M.; Bonaventura, L.; Sacco, R., A semi-Lagrangian discontinuous Galerkin method for scalar advection by incompressible flows, Journal of computational physics, 216, 195-215, (2006) · Zbl 1090.76045
[28] Rossmanith, J.; Seal, D., A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the vlasov – poisson equations, Journal of computational physics, 230, 6203-6232, (2011) · Zbl 1419.76506
[29] Sonnendruecker, E.; Roche, J.; Bertrand, P.; Ghizzo, A., The semi-Lagrangian method for the numerical resolution of the Vlasov equation, Journal of computational physics, 149, 201-220, (1999) · Zbl 0934.76073
[30] Staniforth, A.; Cote, J., Semi-Lagrangian integration schemes for atmospheric models – a review, Monthly weather review, 119, 2206-2223, (1991)
[31] Suli, E., Convergence and nonlinear stability of the lagrange – galerkin method for the navier – stokes equations, Numerische Mathematik, 53, 459-483, (1988) · Zbl 0637.76024
[32] Umeda, T.; Ashour-Abdalla, M.; Schriver, D., Comparison of numerical interpolation schemes for one-dimensional electrostatic Vlasov code, Journal of plasma physics, 72, 1057-1060, (2006)
[33] Wang, H.; Ewing, R.; Qin, G.; Lyons, S.; Al-Lawatia, M.; Man, S., A family of eulerian – lagrangian localized adjoint methods for multi-dimensional advection-reaction equations, Journal of computational physics, 152, 120-163, (1999) · Zbl 0956.76050
[34] Yabe, T.; Xiao, F.; Utsumi, T., The constrained interpolation profile method for multiphase analysis, Journal of computational physics, 169, 556-593, (2001) · Zbl 1047.76104
[35] Zhang, X.; Shu, C.-W., A genuinely high order total variation diminishing scheme for one-dimensional scalar conservation laws, SIAM journal on numerical analysis, 48, 772-795, (2010) · Zbl 1226.65083
[36] Zhang, X.; Shu, C.-W., On maximum-principle-satisfying high order schemes for scalar conservation laws, Journal of computational physics, 229, 3091-3120, (2010) · Zbl 1187.65096
[37] Zhang, X.; Shu, C.-W., On positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes, Journal of computational physics, 229, 8918-8934, (2010) · Zbl 1282.76128
[38] Zhou, T.; Guo, Y.; Shu, C.-W., Numerical study on Landau damping, Physica D: nonlinear phenomena, 157, 322-333, (2001) · Zbl 0972.82083
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