AI 中文总结
本研究提出不依赖显式时间积分的通用阶BUG积分器，经误差界证明和数值基准测试，其性能显著优于现有BUG积分器，可用于动态低秩近似的高阶数值计算。
AI 中文摘要
动态低秩近似已成为众多学科中广泛应用的数值方法，其核心思想是将随时间变化的微分方程的矩阵或张量值解表示为低秩分解形式。该分解的演化会产生高度刚性的动力学特性，因此需要推导不易受这种刚性影响的新型时间积分方法。基更新与伽辽金（BUG）积分器是一类颇具前景的积分器，因为它们支持隐式时间积分并能保持结构特性。然而，当前的BUG积分器仅局限于二阶精度，而通用阶BUG积分器被设计为显式时间积分方法的投影，这严重限制了它们的应用。本研究提出了不依赖显式时间积分方案的通用阶BUG积分器，该积分器在实现高阶精度时所需的基函数数量更少。我们对所提出的增广并行BUG积分器证明了通用阶误差界，并通过一系列刚性和非刚性数值基准测试展示了其性能，结果表明它们显著优于之前的BUG版本。
英文摘要
Dynamical low-rank approximation has become a widely used numerical method in diverse disciplines. Its main idea is to represent the matrix or tensor-valued solution to a time-dependent differential equation as a low-rank factorization. The evolution of the factorization leads to highly stiff dynamics and requires the derivation of novel time integration methods that are not prone to this stiffness. A promising family of integrators are basis-update \& Galerkin (BUG) integrators as they enable implicit time integration and structure--preservation. However, current BUG integrators are limited to second--order accuracy while general-order BUG integrators are designed as projections of explicit time integration methods, thus severely limiting their use. In this work, we propose general-order BUG integrators that do not rely on an explicit time integration scheme while requiring a smaller number of basis functions to achieve high-order accuracy. We prove a general order error bound for the proposed augmented and parallel BUG integrators and demonstrate their behaviour for a series of stiff and non-stiff numerical benchmarks in which they significantly outperform previous BUG versions.
