Multi-Scale Kinetic Simulation: Asymptotic Preserving IMEX-BDF-DG Schemes with Three Implicit-Explicit Partitionings

TL;DR

This paper develops and analyzes three families of asymptotic preserving numerical schemes for multi-scale kinetic transport models, combining DG spatial discretization, discrete ordinates in velocity, and IMEX BDF time integration.

Contribution

It introduces novel AP methods with different IMEX partitionings for kinetic models, enhancing stability, accuracy, and efficiency in multi-scale simulations.

Findings

Methods preserve asymptotic behavior on under-resolved meshes.

Comparison shows differences in stability, accuracy, and computational complexity.

Multi-step IMEX BDF methods outperform some Runge-Kutta schemes in efficiency.

Abstract

Kinetic transport models are mesoscopic mathematical descriptions of the transport of particles as well as their interactions with the background media or among themselves, and they have wide applications in many areas of mathematical physics such as nuclear and biomedical engineering, rarefied gas dynamics, and plasma physics. They are often multi-scale, with different characteristics (e.g. hyperbolic, diffusive) depending on the material properties. As our continuing effort to design and analyze numerical methods for accurate and robust simulation of the multi-scale kinetic transport models, in this work, we consider a linear kinetic transport model, a simplified radiative transfer equation, in a diffusive scaling, and propose and analyze three families of asymptotic preserving (AP) methods. Numerical methods with the AP property, that is to preserve the asymptotic behavior of the models at the discrete level on under-resolved meshes, can work uniformly well to simulate multi-scale models across a wide range of scales. The proposed methods start from the micro-macro decomposition of the model, and involve discontinuous Galerkin (DG) methods in space, the discrete ordinates method (i.e. $S_N$ method) in velocity, and implicit-explicit (IMEX) BDF methods in time, with three different IMEX partitionings. A systematic study, both analytically and computationally, is presented regarding their difference in stability, accuracy, computational complexity and AP property. These methods, with multi-step time integrators, are also compared in terms of their accuracy and efficiency with the ones that only differ in using certain IMEX Runge-Kutta methods in time. Together with our previous developments, the present work further contributes to high order DG AP methods for multi-scale kinetic simulation, especially by utilizing the structure of the micro-macro decomposition of the models.