On the stability of the low-rank projector-splitting integrators for hyperbolic and parabolic equations

TL;DR

This paper analyzes the stability of low-rank projector-splitting integrators applied to hyperbolic and parabolic equations, revealing conditions for stability and improvements via splitting methods.

Contribution

It provides a von Neumann-type stability analysis for PSI methods, comparing discretize-then-project and project-then-discretize formulations under various splitting schemes.

Findings

Stability conditions for DtP and PtD are the same under Lie-Trotter splitting for hyperbolic equations.

Strang splitting can significantly enlarge the stability region.

Unconditional stability for parabolic equations can be achieved with Crank-Nicolson or hybrid schemes.

Abstract

We study the stability of a class of dynamical low-rank methods--the projector-splitting integrator (PSI)--applied to linear hyperbolic and parabolic equations. Using a von Neumann-type analysis, we investigate the stability of such low-rank time integrator coupled with standard spatial discretizations, including upwind and central finite difference schemes, under two commonly used formulations: discretize-then-project (DtP) and project-then-discretize (PtD). For hyperbolic equations, we show that the stability conditions for DtP and PtD are the same under Lie-Trotter splitting, and that the stability region can be significantly enlarged by using Strang splitting. For parabolic equations, despite the presence of a negative S-step, unconditional stability can still be achieved by employing Crank-Nicolson or a hybrid forward-backward Euler scheme in time stepping. While our analysis focuses on simplified model problems, it offers insight into the stability behavior of PSI for more complex systems, such as those arising in kinetic theory.