A robust and conservative dynamical low-rank algorithm

Abstract

Dynamical low-rank approximation, as has been demonstrated recently, can be extremely efficient in solving kinetic equations. However, a major deficiency is that it does not preserve the structure of the underlying physical problem. For example, the classic dynamical low-rank methods violate mass, momentum, and energy conservation. In Einkemmer and Joseph (2021) [9] a conservative dynamical low-rank approach has been proposed. However, directly integrating the resulting equations of motion, similar to the classic dynamical low-rank approach, results in an ill-posed scheme. In this work we propose a robust, i.e. well-posed, low-rank integrator that conserves mass and momentum (up to machine precision) and significantly improves energy conservation. We also report improved qualitative results for some problems and show how the approach can be combined with a rank adaptive scheme.