Asymptotic preserving IMEX-DG-S schemes for linear kinetic transport equations based on Schur complement

We consider a linear kinetic transport equation under a diffusive scaling, that converges to a diffusion equation as the Knudsen number $\varepsilon\rightarrow0$. In [3, 21], to achieve the asymptotic preserving (AP) property and unconditional stability in the diffusive regime with $\varepsilon\ll 1$, numerical schemes are developed based on an additional reformulation of the even-odd or micro-macro decomposed version of the equation. The key of the reformulation is to add a weighted diffusive term on both sides of one equation in the decomposed system. The choice of the weight function, however, is problem-dependent and ad-hoc, and it can affect the performance of numerical simulations. To avoid issues related to the choice of the weight function and still obtain the AP property and unconditional stability in the diffusive regime, we propose in this paper a new family of AP schemes, termed as IMEX-DG-S schemes, directly solving the micro-macro decomposed system without any further reformulation. The main ingredients of the IMEX-DG-S schemes include globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta (RK) temporal discretizations with a new IMEX strategy, discontinuous Galerkin (DG) spatial discretizations, discrete ordinate methods for the velocity space, and the application of the Schur complement to the algebraic form of the schemes to control the overall computational cost. The AP property of the schemes is shown formally. With an energy type stability analysis applied to the first order scheme, and Fourier type stability analysis applied to the first to third order schemes, we confirm the uniform stability of the methods with respect to $\varepsilon$ and the unconditional stability in the diffusive regime. A series of numerical examples are presented to demonstrate the performance of the new schemes.

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

[2]  G. C. Pomraning.  The Equations of Radiation Hydrodynamics , 2005 .

[3]  Tao Xiong,et al.  High order asymptotic preserving DG-IMEX schemes for discrete-velocity kinetic equations in a diffusive scaling , 2013, J. Comput. Phys..

[4]  Bernardo Cockburn,et al.  The Runge-Kutta local projection discontinous Galerkin finite element method for conservation laws , 1990 .

[5]  Bernardo Cockburn,et al.  Optimal a priori error estimates for the hp-version of the local discontinuous Galerkin method for convection-diffusion problems , 2002, Math. Comput..

[6]  Zhichao Peng,et al.  Stability-enhanced AP IMEX-LDG schemes for linear kinetic transport equations under a diffusive scaling , 2020, J. Comput. Phys..

[7]  E. Lewis,et al.  Computational Methods of Neutron Transport , 1993 .

[8]  E. Larsen,et al.  Asymptotic solutions of numerical transport problems in optically thick, diffusive regimes II , 1989 .

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

[10]  Fuzhen Zhang.  The Schur complement and its applications , 2005 .

[11]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[12]  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..

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

[14]  Chi-Wang Shu,et al.  Optimal non-dissipative discontinuous Galerkin methods for Maxwell's equations in Drude metamaterials , 2017, Comput. Math. Appl..

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

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

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

[18]  Barry F. Smith,et al.  PETSc Users Manual , 2019 .

[19]  P. Degond.  Asymptotic-Preserving Schemes for Fluid Models of Plasmas , 2011, 1104.1869.

[20]  Zhichao Peng,et al.  Stability-enhanced AP IMEX1-LDG method: energy-based stability and rigorous AP property , 2020, ArXiv.

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

[22]  Ryan G. McClarren,et al.  The effects of slope limiting on asymptotic-preserving numerical methods for hyperbolic conservation laws , 2008, J. Comput. Phys..

[23]  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..

[24]  Giovanni Russo,et al.  Uniformly Accurate Schemes for Hyperbolic Systems with Relaxation , 1997 .

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

[26]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

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