A Galerkin Alternating Projection Method for Kinetic Equations in the Diffusive Limit

TL;DR

This paper introduces a novel Galerkin Alternating Projection scheme within the DLRA framework for high-dimensional kinetic equations, proving its accuracy, asymptotic-preserving property, and efficiency across regimes with numerical validation.

Contribution

The paper presents a new integrator that is asymptotic-preserving, high-order, and free of CFL constraints in the diffusive limit, advancing numerical methods for kinetic equations.

Findings

GAP scheme is asymptotic-preserving for RTE.

GAP achieves high-order accuracy in the diffusive limit.

Numerical experiments confirm robustness and efficiency.

Abstract

The numerical approximation of high-dimensional evolution equations poses significant computational challenges, particularly in kinetic theory and radiative transfer. In this work, we introduce the Galerkin Alternating Projection (GAP) scheme, a novel integrator derived within the Dynamical Low-Rank Approximation (DLRA) framework. We perform a rigorous error analysis, establishing local and global accuracy using standard ODE techniques. Furthermore, we prove that GAP possesses the Asymptotic-Preserving (AP) property when applied to the Radiative Transfer Equation (RTE), ensuring consistent behavior across both kinetic and diffusive regimes. In the diffusive regime, the K-step of the GAP integrator directly becomes the limit equation. In particular, this means that we can easily obtain schemes that even in the diffusive regime are free of a CFL condition, do not require well prepared initial data, and can have arbitrary order in the diffusive limit (in contrast to the semi-implicit and implicit schemes available in the literature). Numerical experiments support the theoretical findings and demonstrate the robustness and efficiency of the proposed method.