Stable and asymptotic preserving space-time discretizations of a linear kinetic transport equation in diffusive scaling

TL;DR

This paper introduces an energy-stable space-time discretization method for a linear kinetic transport equation, ensuring stability and asymptotic preservation in diffusive scaling through micro-macro decomposition.

Contribution

It presents a novel unconditionally energy-stable tensor-product discretization framework with a new boundary treatment, guaranteeing stability and asymptotic preservation.

Findings

Numerical results confirm convergence for smooth problems.

The boundary treatment is energy stable.

The method preserves asymptotic limits at the discrete level.

Abstract

We develop an unconditionally energy-stable tensor-product space-time discretization framework for the solution of a linear kinetic transport equation in one space dimension. The kinetic equation is a simplified model of radiative transfer formulated as a hyperbolic balance law in diffusive scaling for a particle distribution function of the independent variables space, time and velocity. Our numerical discretization is based on the well-known technique of micro-macro decomposition which results in a system of balance laws for equilibrium and non-equilibrium quantities and facilitates preservation of the asymptotic limit for vanishing scaling parameters at the discrete level. We prove fully discrete stability and asymptotic preservation for general spatial and temporal discretizations having the summation-by-parts property. A new provably energy-stable Dirichlet boundary treatment for the micro-macro decomposed system is developed based on the introduction of simultaneous approximation terms. Numerical results show convergence for smooth problems and demonstrate energy stability of the proposed boundary treatment.