Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations

TL;DR

This paper analyzes the stability of fully discrete projector-splitting schemes for dynamical low-rank approximation of random parabolic equations, establishing conditions for explicit, semi-implicit, and implicit schemes, with numerical validation.

Contribution

It introduces and analyzes a class of stable, projector-splitting numerical schemes for DLR approximation of random parabolic equations, including conditions for stability and connections to existing methods.

Findings

Explicit and semi-implicit schemes are conditionally stable under a CFL condition.

Implicit schemes are unconditionally stable.

Semi-implicit scheme can be unconditionally stable with small randomness.

Abstract

We consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a parabolic type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. Furthermore, we show that these schemes can be interpreted as projector-splitting integrators and are strongly related to the scheme proposed by Lubich et al. [BIT Num. Math., 54:171-188, 2014; SIAM J. on Num. Anal., 53:917-941, 2015], to which our stability analysis applies as well. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions.