High order asymptotic preserving DG-IMEX schemes for discrete-velocity kinetic equations in a diffusive scaling

In this paper, we develop a family of high order asymptotic preserving schemes for some discrete-velocity kinetic equations under a diffusive scaling, that in the asymptotic limit lead to macroscopic models such as the heat equation, the porous media equation, the advection-diffusion equation, and the viscous Burgers' equation. Our approach is based on the micro-macro reformulation of the kinetic equation which involves a natural decomposition of the equation to the equilibrium and non-equilibrium parts. To achieve high order accuracy and uniform stability as well as to capture the correct asymptotic limit, two new ingredients are employed in the proposed methods: discontinuous Galerkin (DG) spatial discretization of arbitrary order of accuracy with suitable numerical fluxes; high order globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta scheme in time equipped with a properly chosen implicit-explicit strategy. Formal asymptotic analysis shows that the proposed scheme in the limit of e ? 0 is a consistent high order discretization for the limiting equation. Numerical results are presented to demonstrate the stability and high order accuracy of the proposed schemes together with their performance in the limit. Our methods are also tested for the continuous-velocity one-group transport equation in slab geometry and for several examples with spatially varying parameters.

[1]  Laure Saint-Raymond,et al.  Hydrodynamic Limits of the Boltzmann Equation , 2009 .

[2]  Lorenzo Pareschi,et al.  Efficient Asymptotic Preserving Deterministic methods for the Boltzmann Equation , 2011 .

[3]  J. Hesthaven,et al.  Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications , 2007 .

[4]  Giovanni Russo,et al.  On a Class of Uniformly Accurate IMEX Runge--Kutta Schemes and Applications to Hyperbolic Systems with Relaxation , 2009, SIAM J. Sci. Comput..

[5]  SEBASTIANO BOSCARINO.  Error Analysis of IMEX Runge-Kutta Methods Derived from Differential-Algebraic Systems , 2007, SIAM J. Numer. Anal..

[6]  Frédéric Nataf,et al.  Robin Schwarz Algorithm for the NICEM Method: The $\mathbf{P}_q$ Finite Element Case , 2014, SIAM J. Numer. Anal..

[7]  Lorenzo Pareschi,et al.  Numerical schemes for kinetic equations in diffusive regimes , 1998 .

[8]  Chi-Wang Shu,et al.  Local Discontinuous Galerkin Methods for High-Order Time-Dependent Partial Differential Equations , 2009 .

[9]  François Golse,et al.  Diffusion approximation and entropy-based moment closure for kinetic equations , 2005 .

[10]  Jean-Luc Guermond,et al.  Asymptotic Analysis of Upwind Discontinuous Galerkin Approximation of the Radiative Transport Equation in the Diffusive Limit , 2010, SIAM J. Numer. Anal..

[11]  F. Golse,et al.  Fluid dynamic limits of kinetic equations. I. Formal derivations , 1991 .

[12]  Steven J. Ruuth,et al.  Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations , 1997 .

[13]  Tai-Ping Liu,et al.  Boltzmann Equation: Micro-Macro Decompositions and Positivity of Shock Profiles , 2004 .

[14]  Mohammed Lemou,et al.  Micro-Macro Schemes for Kinetic Equations Including Boundary Layers , 2012, SIAM J. Sci. Comput..

[15]  Shi Jin,et al.  Uniformly Accurate Diffusive Relaxation Schemes for Multiscale Transport Equations , 2000, SIAM J. Numer. Anal..

[16]  C. D. Levermore,et al.  Numerical Schemes for Hyperbolic Conservation Laws with Stiff Relaxation Terms , 1996 .

[17]  Shi Jin.  ASYMPTOTIC PRESERVING (AP) SCHEMES FOR MULTISCALE KINETIC AND HYPERBOLIC EQUATIONS: A REVIEW , 2010 .

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

[19]  Bengt Fornberg,et al.  Accurate numerical resolution of transients in initial-boundary value problems for the heat equation , 2003 .

[20]  Luc Mieussens,et al.  A New Asymptotic Preserving Scheme Based on Micro-Macro Formulation for Linear Kinetic Equations in the Diffusion Limit , 2008, SIAM J. Sci. Comput..

[21]  Tai Tsun Wu,et al.  A completely solvable model of the nonlinear Boltzmann equation , 1982 .

[22]  Shi Jin,et al.  A Smooth Transition Model between Kinetic and Diffusion Equations , 2004, SIAM J. Numer. Anal..

[23]  Chi-Wang Shu,et al.  On a cell entropy inequality for discontinuous Galerkin methods , 1994 .

[24]  A. Klar.  An Asymptotic-Induced Scheme for Nonstationary Transport Equations in the Diffusive Limit , 1998 .

[25]  Lorenzo Pareschi,et al.  Implicit-Explicit Runge-Kutta Schemes for Hyperbolic Systems and Kinetic Equations in the Diffusion Limit , 2013, SIAM J. Sci. Comput..

[26]  Tao Xiong,et al.  Analysis of Asymptotic Preserving DG-IMEX Schemes for Linear Kinetic Transport Equations in a Diffusive Scaling , 2013, SIAM J. Numer. Anal..

[27]  G. Russo,et al.  Implicit-explicit runge-kutta schemes and applications to hyperbolic systems with relaxation , 2005 .

[28]  Lorenzo Pareschi,et al.  Diffusive Relaxation Schemes for Multiscale Discrete-Velocity Kinetic Equations , 1998 .

[29]  Patrick Le Tallec,et al.  Coupling Boltzmann and Euler equations without overlapping , 1992 .

[30]  Giovanni Russo,et al.  Flux-Explicit IMEX Runge-Kutta Schemes for Hyperbolic to Parabolic Relaxation Problems , 2013, SIAM J. Numer. Anal..

[31]  C. D. Levermore,et al.  Moment closure hierarchies for kinetic theories , 1996 .

[32]  Thierry Goudon,et al.  Numerical Schemes of Diffusion Asymptotics and Moment Closures for Kinetic Equations , 2008, J. Sci. Comput..

[33]  Luc Mieussens,et al.  Analysis of an Asymptotic Preserving Scheme for Linear Kinetic Equations in the Diffusion Limit , 2009, SIAM J. Numer. Anal..

[34]  Béatrice Rivière,et al.  Discontinuous Galerkin methods for solving elliptic and parabolic equations - theory and implementation , 2008, Frontiers in applied mathematics.

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