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.

翻译：动力系统低秩近似已成为对规模过大的矩阵微分方程进行在线模型降阶的重要工具。其核心在于构建对微小奇异值及由此产生的低秩矩阵表示中正交因子大时间导数具有鲁棒性的积分器。近年来，鲁棒性基更新与伽辽金（BUG）类积分器被提出。这些方法不需要对解进行时间反向演化，常具有优越的结构保持特性，并允许低秩因子的并行时间更新。BUG框架具有足够灵活性以适应这些及更多需求。然而，目前提出的BUG方法仅具有一阶鲁棒误差界。本工作基于中点法则提出了一种用于动力系统低秩近似的二阶BUG积分器。该积分器首先使用一阶BUG积分器执行半步更新，随后通过适当增广基进行伽辽金更新。我们证明了鲁棒的二阶误差界，该误差界还显示了对向量场法向分量的依赖性的改进。这些严格结果通过一系列数值实验得到验证和补充。