Rank-adaptive dynamical low-rank integrators for first-order and second-order matrix differential equations

Dynamical low-rank integrators for first-order problems have been studied extensively since the 2010s. Recently, dynamical low-rank integrators for second-order problems have been developed in [10]. In this paper, we propose a novel strategy for choosing the rank adaptively, which is applicable for integrators of first and second-order equations. Our adaptive algorithms are based on a combination of error estimators for the local time-discretization error and for the low-rank error. Numerical experiments illustrate the performance of the new integrators.