Convergence of a low-rank Lie--Trotter splitting for stiff matrix differential equations

Abstract: We propose a numerical integrator for determining low-rank approximations to solutions of large-scale matrix differential equations. The considered differential equations are semilinear and stiff. Our method consists of first splitting the differential equation into a stiff and a non-stiff part, respectively, and then following a dynamical low-rank approach. We conduct an error analysis of the proposed procedure, which is independent of the stiffness and robust with respect to possibly small singular values in the approximation matrix. Following the proposed method, we show how to obtain low-rank approximations for differential Lyapunov and for differential Riccati equations. Our theory is illustrated by numerical experiments.