On the stability of the low-rank projector-splitting integrators for hyperbolic and parabolic equations
This work provides theoretical stability guarantees for a widely used low-rank time integrator, benefiting researchers applying dynamical low-rank methods to hyperbolic and parabolic PDEs, though the analysis is limited to simplified model problems.
The paper analyzes the stability of the projector-splitting integrator (PSI) for low-rank approximations of linear hyperbolic and parabolic equations. It finds that stability conditions for discretize-then-project and project-then-discretize formulations are identical under Lie-Trotter splitting, and that Strang splitting enlarges the stability region for hyperbolic equations, while unconditional stability for parabolic equations is achievable with Crank-Nicolson or hybrid forward-backward Euler schemes.
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.