Conservative Multi-dimensional Semi-Lagrangian Finite Difference Scheme: Stability and Applications to the Kinetic and Fluid Simulations

In this paper, we propose a mass conservative semi-Lagrangian finite difference scheme for multi-dimensional problems without dimensional splitting. The semi-Lagrangian scheme, based on tracing characteristics backward in time from grid points, does not necessarily conserve the total mass. To ensure mass conservation, we propose a conservative correction procedure based on a flux difference form. Such procedure guarantees local mass conservation, while introducing time step constraints for stability. We theoretically investigate such stability constraints from an ODE point of view by assuming exact evaluation of spatial differential operators and from the Fourier analysis for linear PDEs. The scheme is tested by classical two dimensional linear passive-transport problems, such as linear advection, rotation and swirling deformation. The scheme is applied to solve the nonlinear Vlasov–Poisson system and guiding center Vlasov model using high order tracing schemes. The effectiveness of the proposed conservative semi-Lagrangian scheme is demonstrated numerically by extensive numerical tests.

[1]  José A. Carrillo,et al.  Nonoscillatory Interpolation Methods Applied to Vlasov-Based Models , 2007, SIAM J. Sci. Comput..

[2]  Chi-Wang Shu,et al.  High Order Weighted Essentially Nonoscillatory Schemes for Convection Dominated Problems , 2009, SIAM Rev..

[3]  Chi-Wang Shu,et al.  Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov-Poisson system , 2011, J. Comput. Phys..

[4]  Andrew J. Christlieb,et al.  A conservative high order semi-Lagrangian WENO method for the Vlasov equation , 2010, J. Comput. Phys..

[5]  E. Sonnendrücker,et al.  The Semi-Lagrangian Method for the Numerical Resolution of the Vlasov Equation , 1999 .

[6]  O. Pironneau.  On the transport-diffusion algorithm and its applications to the Navier-Stokes equations , 1982 .

[7]  Endre Süli,et al.  Stability of the Lagrange-Galerkin method with non-exact integration , 1988 .

[8]  Wei Guo,et al.  A high order time splitting method based on integral deferred correction for semi-Lagrangian Vlasov simulations , 2014, J. Comput. Phys..

[9]  Olivier Coulaud,et al.  Instability of the time splitting scheme for the one-dimensional and relativistic Vlasov-Maxwell system , 2003 .

[10]  George Em Karniadakis,et al.  A semi-Lagrangian high-order method for Navier-Stokes equations , 2001 .

[11]  Frank Losasso,et al.  A fast and accurate semi-Lagrangian particle level set method , 2005 .

[12]  Chi-Wang Shu,et al.  Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow , 2011, J. Comput. Phys..

[13]  Shian‐Jiann Lin,et al.  Multidimensional Flux-Form Semi-Lagrangian Transport Schemes , 1996 .

[14]  Wei Guo,et al.  A High Order Conservative Semi-Lagrangian Discontinuous Galerkin Method for Two-Dimensional Transport Simulations , 2016, Journal of Scientific Computing.

[15]  Eric Sonnendrücker,et al.  Conservative semi-Lagrangian schemes for Vlasov equations , 2010, J. Comput. Phys..

[16]  Jing-Mei Qiu,et al.  A Conservative Semi-Lagrangian Discontinuous Galerkin Scheme on the Cubed Sphere , 2014 .

[17]  S. Hirstoaga,et al.  Long Time Behaviour of an Exponential Integrator for a Vlasov-Poisson System with Strong Magnetic Field , 2015 .

[18]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[19]  G. Knorr,et al.  The integration of the vlasov equation in configuration space , 1976 .

[20]  Magdi Shoucri.  A two-level implicit scheme for the numerical solution of the linearized vorticity equation , 1981 .

[21]  David C. Seal,et al.  A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov-Poisson equations , 2010, J. Comput. Phys..

[22]  Tao Xiong,et al.  High Order Multi-dimensional Characteristics Tracing for the Incompressible Euler Equation and the Guiding-Center Vlasov Equation , 2018, J. Sci. Comput..

[23]  Chang Yang,et al.  Conservative and non-conservative methods based on Hermite weighted essentially non-oscillatory reconstruction for Vlasov equations , 2013, J. Comput. Phys..

[24]  Wei Guo,et al.  Hybrid semi-Lagrangian finite element-finite difference methods for the Vlasov equation , 2013, J. Comput. Phys..

[25]  Wei Guo,et al.  A high order semi-Lagrangian discontinuous Galerkin method for Vlasov-Poisson simulations without operator splitting , 2017, J. Comput. Phys..

[26]  P. Bertrand,et al.  Conservative numerical schemes for the Vlasov equation , 2001 .

[27]  A. Staniforth,et al.  Semi-Lagrangian integration schemes for atmospheric models - A review , 1991 .

[28]  Giovanni Russo,et al.  A High Order Multi-Dimensional Characteristic Tracing Strategy for the Vlasov–Poisson System , 2017, J. Sci. Comput..

[29]  J. Strain.  Semi-Lagrangian Methods for Level Set Equations , 1999 .

[30]  R. Glassey,et al.  The Cauchy Problem in Kinetic Theory , 1987 .