Stability-enhanced AP IMEX1-LDG method: energy-based stability and rigorous AP property

In our recent work [22], a family of high order asymptotic preserving (AP) methods, termed as IMEX-LDG methods, are designed to solve some linear kinetic transport equations, including the one-group transport equation in slab geometry and the telegraph equation, in a diffusive scaling. As the Knudsen number $\varepsilon$ goes to zero, the limiting schemes are implicit discretizations to the limiting diffusive equation. Both Fourier analysis and numerical experiments imply the methods are unconditionally stable in the diffusive regime when $\varepsilon\ll1$. In this paper, we develop an energy approach to establish the numerical stability of the IMEX1-LDG method, the sub-family of the methods that is first order accurate in time and arbitrary order in space, for the model with general material properties. Our analysis is the first to simultaneously confirm unconditional stability when $\varepsilon\ll1$ and the uniform stability property with respect to $\varepsilon$. To capture the unconditional stability, a novel discrete energy is introduced by better exploring the contribution of the scattering term in different regimes. A general form of the weight function, introduced to obtain the unconditional stability for $\varepsilon\ll1$, is also for the first time considered in such stability analysis. Based on the uniform stability, a rigorous asymptotic analysis is then carried out to show the AP property.

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

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

[3]  Jingwei Hu,et al.  On the uniform accuracy of implicit-explicit backward differentiation formulas (IMEX-BDF) for stiff hyperbolic relaxation systems and kinetic equations , 2019, Math. Comput..

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

[5]  Erwan Faou,et al.  Analysis of an Asymptotic Preserving Scheme for Stochastic Linear Kinetic Equations in the Diffusion Limit , 2019, SIAM/ASA J. Uncertain. Quantification.

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

[7]  Luc Mieussens,et al.  On the asymptotic preserving property of the unified gas kinetic scheme for the diffusion limit of linear kinetic models , 2013, J. Comput. Phys..

[8]  N. Crouseilles,et al.  An asymptotic preserving scheme based on a micro-macro decomposition for collisional Vlasov equations: diffusion and high-field scaling limits. , 2011 .

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

[10]  Francis Filbet,et al.  Analysis of an Asymptotic Preserving Scheme for Relaxation Systems , 2011 .

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

[12]  Giacomo Dimarco,et al.  Asymptotically complexity diminishing schemes (ACDS) for kinetic equations in the diffusive scaling , 2019, J. Comput. Phys..

[13]  François Golse,et al.  The Convergence of Numerical Transfer Schemes in Diffusive Regimes I: Discrete-Ordinate Method , 1999 .

[14]  Axel Klar,et al.  Uniform Stability of a Finite Difference Scheme for Transport Equations in Diffusive Regimes , 2002, SIAM J. Numer. Anal..

[15]  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..

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

[17]  Philippe G. LeFloch,et al.  High-Order Asymptotic-Preserving Methods for Fully Nonlinear Relaxation Problems , 2012, SIAM J. Sci. Comput..

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

[19]  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..

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

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

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

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

[24]  G. Dimarco,et al.  Implicit-Explicit Runge-Kutta schemes for the Boltzmann-Poisson system for semiconductors , 2013, 1305.1759.

[25]  Fengyan Li,et al.  Operator Bounds and Time Step Conditions for the DG and Central DG Methods , 2015, J. Sci. Comput..

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

[27]  Maxime Herda,et al.  Hypocoercivity and diffusion limit of a finite volume scheme for linear kinetic equations , 2018, Math. Comput..

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

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