A robust second-order low-rank BUG integrator based on the midpoint rule

Abstract

Dynamical low-rank approximation has become a valuable tool to perform an on-the-fly model order reduction for prohibitively large matrix differential equations. A core ingredient is the construction of integrators that are robust to the presence of small singular values and the resulting large time derivatives of the orthogonal factors in the low-rank matrix representation. Recently, the robust basis-update & Galerkin (BUG) class of integrators has been introduced. These methods require no steps that evolve the solution backward in time, often have favourable structure-preserving properties, and allow for parallel time-updates of the low-rank factors. The BUG framework is flexible enough to allow for adaptations to these and further requirements. However, the BUG methods presented so far have only first-order robust error bounds. This work proposes a second-order BUG integrator for dynamical low-rank approximation based on the midpoint quadrature rule. The integrator first performs a half-step with a first-order BUG integrator, followed by a Galerkin update with a suitably augmented basis. We prove a robust second-order error bound which in addition shows an improved dependence on the normal component of the vector field. These rigorous results are illustrated and complemented by a number of numerical experiments.