A semi-implicit dynamical low-rank discontinuous Galerkin method for space homogeneous kinetic equations. Part I: emission and absorption

TL;DR

This paper introduces a semi-implicit dynamical low-rank discontinuous Galerkin method tailored for space homogeneous kinetic equations, effectively reducing computational costs while ensuring convergence to equilibrium.

Contribution

It develops a novel semi-implicit DLR-DG scheme that maintains consistency with classical DG methods and proves convergence properties for kinetic equations with emission and absorption.

Findings

The DLR-DG method is well-posed and converges to equilibrium.

Numerical results confirm theoretical convergence and efficiency.

The approach reduces computational costs for high-dimensional kinetic problems.

Abstract

Dynamical low-rank approximation (DLRA) is an emerging tool for reducing computational costs and provides memory savings when solving high-dimensional problems. In this work, we propose and analyze a semi-implicit dynamical low-rank discontinuous Galerkin (DLR-DG) method for the space homogeneous kinetic equation with a relaxation operator, modeling the emission and absorption of particles by a background medium. Both DLRA and the DG scheme can be formulated as Galerkin equations. To ensure their consistency, a weighted DLRA is introduced so that the resulting DLR-DG solution is a solution to the fully discrete DG scheme in a subspace of the classical DG solution space. Similar to the classical DG method, we show that the proposed DLR-DG method is well-posed. We also identify conditions such that the DLR-DG solution converges to the equilibrium. Numerical results are presented to demonstrate the theoretical findings.