On the stability of robust dynamical low-rank approximations for hyperbolic problems

TL;DR

This paper analyzes the stability of dynamical low-rank approximations for hyperbolic problems, identifies sources of instability, and proposes improved integrators with better stability properties, supported by numerical experiments.

Contribution

It provides an $L^2$ stability analysis of DLRA methods, introduces a new projector splitting integrator that recovers the CFL condition, and offers a stable low-rank update for kinetic transport.

Findings

The projector splitting integrator's instability is linked to discretization order.

The new integrator based on continuous systems recovers the CFL condition.

Numerical experiments confirm the stability analysis and effectiveness of proposed methods.

Abstract

The dynamical low-rank approximation (DLRA) is used to treat high-dimensional problems that arise in such diverse fields as kinetic transport and uncertainty quantification. Even though it is well known that certain spatial and temporal discretizations when combined with the DLRA approach can result in numerical instability, this phenomenon is poorly understood. In this paper we perform a $L^2$ stability analysis for the corresponding nonlinear equations of motion. This reveals the source of the instability for the projector splitting integrator when first discretizing the equations and then applying the DLRA. Based on this we propose a projector splitting integrator, based on applying DLRA to the continuous system before performing the discretization, that recovers the classic CFL condition. We also show that the unconventional integrator has more favorable stability properties and explain why the projector splitting integrator performs better when approximating higher moments, while the unconventional integrator is generally superior for first order moments. Furthermore, an efficient and stable dynamical low-rank update for the scattering term in kinetic transport is proposed. Numerical experiments for kinetic transport and uncertainty quantification, which confirm the results of the stability analysis, are presented.