High Order Semi-Lagrangian Discontinuous Galerkin Method Coupled with Runge-Kutta Exponential Integrators for Nonlinear Vlasov Dynamics

In this paper, we propose a semi-Lagrangian discontinuous Galerkin method coupled with Runge-Kutta exponential integrators (SLDG-RKEI) for nonlinear Vlasov dynamics. The commutator-free Runge-Kutta (RK) exponential integrators (EI) were proposed by Celledoni, et al. (FGCS, 2003). In the nonlinear transport setting, the RKEI can be used to decompose the evolution of the nonlinear transport into a composition of a sequence of linearized dynamics. The resulting linearized transport equations can be solved by the semi-Lagrangian (SL) discontinuous Galerkin (DG) method proposed in Cai, et al. (JSC, 2017). The proposed method can achieve high order spatial accuracy via the SLDG framework, and high order temporal accuracy via the RK EI. Due to the SL nature, the proposed SLDG-RKEI method is not subject to the CFL condition, thus they have the potential in using larger time-stepping sizes than those in the Eulerian approach. Inheriting advantages from the SLDG method, the proposed SLDG-RKEI schemes are mass conservative, positivity-preserving, have no dimensional splitting error, perform well in resolving complex solution structures, and can be evolved with adaptive time-stepping sizes. We show the performance of the SLDG-RKEI algorithm by classical test problems for the nonlinear Vlasov-Poisson system, as well as the Guiding center Vlasov model. Though that it is not our focus of this paper to explore the SLDG-RKEI scheme for nonlinear hyperbolic conservation laws that develop shocks, we show some preliminary results on schemes' performance on the Burgers' equation.

[1]  Lukas Einkemmer,et al.  Convergence Analysis of a Discontinuous Galerkin/Strang Splitting Approximation for the Vlasov-Poisson Equations , 2012, SIAM J. Numer. Anal..

[2]  Todd Arbogast,et al.  A semi-Lagrangian finite difference WENO scheme for scalar nonlinear conservation laws , 2016, J. Comput. Phys..

[3]  Riccardo Sacco,et al.  A semi-Lagrangian discontinuous Galerkin method for scalar advection by incompressible flows , 2006, J. Comput. Phys..

[4]  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..

[5]  Paul A. Ullrich,et al.  A conservative semi-Lagrangian multi-tracer transport scheme (CSLAM) on the cubed-sphere grid , 2010, J. Comput. Phys..

[6]  Xiangxiong Zhang,et al.  On maximum-principle-satisfying high order schemes for scalar conservation laws , 2010, J. Comput. Phys..

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

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

[9]  Xiaozhou Li,et al.  Smoothness-Increasing Accuracy-Conserving (SIAC) filters for derivative approximations of discontinuous Galerkin (DG) solutions over nonuniform meshes and near boundaries , 2016, J. Comput. Appl. Math..

[10]  Lukas Einkemmer,et al.  A performance comparison of semi-Lagrangian discontinuous Galerkin and spline based Vlasov solvers in four dimensions , 2018, J. Comput. Phys..

[11]  Andrew M. Bradley,et al.  Conservative Multimoment Transport along Characteristics for Discontinuous Galerkin Methods , 2019, SIAM J. Sci. Comput..

[12]  D. Esteve,et al.  A 5D gyrokinetic full-f global semi-Lagrangian code for flux-driven ion turbulence simulations , 2016, Comput. Phys. Commun..

[13]  Hong Wang,et al.  An ELLAM Scheme for Advection-Diffusion Equations in Two Dimensions , 1999, SIAM J. Sci. Comput..

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

[15]  Jennifer K. Ryan,et al.  Extension of a Post Processing Technique for the Discontinuous Galerkin Method for Hyperbolic Equations with Application to an Aeroacoustic Problem , 2005, SIAM J. Sci. Comput..

[16]  Elena Celledoni,et al.  Commutator-free Lie group methods , 2003, Future Gener. Comput. Syst..

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

[18]  Lorenzo Pareschi,et al.  Modeling and Computational Methods for Kinetic Equations , 2012 .

[19]  Raphaël Loubère,et al.  A multiscale fast semi-Lagrangian method for rarefied gas dynamics , 2015, J. Comput. Phys..

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

[21]  Ilaria Perugia,et al.  An A Priori Error Analysis of the Local Discontinuous Galerkin Method for Elliptic Problems , 2000, SIAM J. Numer. Anal..

[22]  Hong Wang,et al.  An Eulerian-Lagrangian Solution Technique for Single-Phase Compositional Flow in Three-Dimensional Porous Media , 2006, Comput. Math. Appl..

[23]  Todd Arbogast,et al.  An Eulerian-Lagrangian WENO finite volume scheme for advection problems , 2012, J. Comput. Phys..

[24]  Nicolas Crouseilles,et al.  DISCONTINUOUS GALERKIN SEMI-LAGRANGIAN METHOD FOR VLASOV-POISSON , 2011 .

[25]  Jianxian Qiu,et al.  An h-Adaptive RKDG Method for the Vlasov–Poisson System , 2016, J. Sci. Comput..

[26]  D. Lee,et al.  A high order characteristic discontinuous Galerkin scheme for advection on unstructured meshes , 2016, J. Comput. Phys..

[27]  Wei Guo,et al.  A High Order Semi-Lagrangian Discontinuous Galerkin Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model Without Operator Splitting , 2018, Journal of Scientific Computing.

[28]  Martin Schreiber,et al.  Semi-Lagrangian Exponential Integration with Application to the Rotating Shallow Water Equations , 2019, SIAM J. Sci. Comput..

[29]  Elena Celledoni,et al.  Semi-Lagrangian Runge-Kutta Exponential Integrators for Convection Dominated Problems , 2009, J. Sci. Comput..

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

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

[32]  B. K. Kometa.  On Semi-Lagrangian Exponential Integrators and Discontinuous Galerkin Methods , 2011 .

[33]  Giovanni Russo,et al.  Boundary conditions for semi-Lagrangian methods for the BGK model , 2016 .

[34]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

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

[36]  Roberto Ferretti,et al.  A fully semi-Lagrangian discretization for the 2D incompressible Navier-Stokes equations in the vorticity-streamfunction formulation , 2018, Appl. Math. Comput..

[37]  Endre Süli,et al.  Enhanced accuracy by post-processing for finite element methods for hyperbolic equations , 2003, Math. Comput..

[38]  K. Kormann,et al.  Stable Interpolation with Isotropic and Anisotropic Gaussians Using Hermite Generating Function , 2019, SIAM J. Sci. Comput..

[39]  Aijie Cheng,et al.  A preliminary study on multiscale ELLAM schemes for transient advection‐diffusion equations , 2010 .

[40]  T. F. Russell,et al.  An Eulerian-Lagrangian localized adjoint method for the advection-diffusion equation , 1990 .

[41]  Francis Filbet,et al.  Vlasov simulations of beams with a moving grid , 2004, Comput. Phys. Commun..

[42]  Elena Celledoni,et al.  High Order Semi-Lagrangian Methods for the Incompressible Navier–Stokes Equations , 2012, J. Sci. Comput..

[43]  Jing-Mei Qiu,et al.  Optimal convergence and superconvergence of semi-Lagrangian discontinuous Galerkin methods for linear convection equations in one space dimension , 2020, Math. Comput..

[44]  S. Hirstoaga,et al.  A new fully two-dimensional conservative semi-Lagrangian method: applications on polar grids, from diocotron instability to ITG turbulence , 2014, The European Physical Journal D.

[45]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[46]  Shian-Jiann Lin,et al.  An explicit flux‐form semi‐lagrangian shallow‐water model on the sphere , 1997 .

[47]  Pierre Bertrand,et al.  A wavelet-MRA-based adaptive semi-Lagrangian method for the relativistic Vlasov-Maxwell system , 2008, J. Comput. Phys..

[48]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

[49]  N. Besse,et al.  Adaptive multiresolution semi-Lagrangian discontinuous Galerkin methods for the Vlasov equations , 2017, J. Comput. Phys..

[50]  Wei Guo,et al.  Comparison of Semi-Lagrangian Discontinuous Galerkin Schemes for Linear and Nonlinear Transport Simulations , 2019, Communications on Applied Mathematics and Computation.

[51]  Jianxian Qiu,et al.  An h-Adaptive RKDG Method for the Two-Dimensional Incompressible Euler Equations and the Guiding Center Vlasov Model , 2017, J. Sci. Comput..

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

[53]  Giovanni Tumolo,et al.  A semi-implicit, semi-Lagrangian, p-adaptive discontinuous Galerkin method for the shallow water equations , 2013, J. Comput. Phys..

[54]  Zhengfu Xu,et al.  High order maximum principle preserving semi-Lagrangian finite difference WENO schemes for the Vlasov equation , 2013, J. Comput. Phys..

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