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 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.
Institute of Fundamental Sciences, Massey University（马斯特里赫特大学基础科学研究所）；Mathematical Sciences, Chalmers University of Technology（查尔姆斯理工大学数学科学系）；Mathematics and Mathematical Statistics, Umeå Universitet（于默奥大学数学与数理统计系）
A decoupled, stable, and second-order integrator for the Landau--Lifshitz--Gilbert equation with magnetoelastic effects
Institute of Information Theory and Automation, Czech Academy of Sciences（自动化与信息理论研究所，捷克科学院）；Department of Mathematics and Statistics, University of Strathclyde（斯特拉斯克莱德大学数学与统计系）；Department of Mathematics, University of Bologna（博洛尼亚大学数学系）
