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

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.