Construction of high-order conservative basis-update and Galerkin dynamical low-rank integrators

TL;DR

This paper develops high-order conservative basis-update and Galerkin dynamical low-rank integrators that preserve invariants and conservation laws in kinetic simulations, improving efficiency and robustness.

Contribution

It introduces extensions to existing DLRA integrators that ensure local conservation laws with minimal computational overhead.

Findings

Successfully preserves invariants in numerical simulations.

Achieves high-order accuracy in kinetic problems.

Requires only minor modifications to existing methods.

Abstract

Numerical simulations of kinetic problems can become prohibitively expensive due to their large memory requirements and computational costs. A method that has proven to successfully reduce these costs is the dynamical low-rank approximation (DLRA). A major accomplishment in the field of DLRA has been the derivation of robust time integrators that are not limited by the stiffness of the DLRA evolution equations. One key question is whether such robust time integrators can be made locally conservative, i.e., can they preserve the invariants and associated conservation laws of the original problem? In this work, we propose extensions to commonly used basis-update \& Galerkin (BUG) integrators that preserve invariants of the solution as well as the associated conservation laws with little or no additional computational cost. This approach requires only minor modifications of existing implementations. The properties of these integrators are investigated by performing numerical simulations in radiation transport and plasma physics.