In this paper, we develop a low-rank method with high-order temporal accuracy using spectral deferred correction (SDC) to compute linear matrix differential equations. In [1], a low rank numerical method is proposed to correct the modeling error of the basis update and the Galerkin (BUG) method, which is a computational approach for DLRA. This method (merge-BUG/mBUG method) has been demonstrated to be first order convergent for general advection-diffusion problems. In this paper, we explore using SDC to elevate the convergence order of the mBUG method. In SDC, we start by computing a first-order solution by mBUG, and then perform successive updates by computing low-rank solutions to the Picard integral equation. Rather than a straightforward application of SDC with mBUG, we propose two aspects to improve computational efficiency. The first is to reduce the intermediate numerical rank by detailed analysis of dependence of truncation parameter on the correction levels. The second aspect is a careful choice of subspaces in the successive correction to avoid inverting large linear systems (from the K- and L-steps in BUG). We prove that the resulting scheme is high-order accurate for the Lipschitz continuous and bounded dynamical system. We consider numerical rank control in our framework by comparing two low-rank truncation strategies: the hard truncation strategy by truncated singular value decomposition and the soft truncation strategy by soft thresholding. We demonstrate numerically that soft thresholding offers better rank control in particular for higher-order schemes for weakly (or non-)dissipative problems.

翻译：本文提出了一种结合谱延迟校正(SDC)的高时间精度低秩方法，用于计算线性矩阵微分方程。文献[1]提出了一种低秩数值方法（合并基更新与伽辽金法/改进BUG方法），用于修正动态低秩逼近(DLRA)中基更新与伽辽金(BUG)方法的建模误差。该方法已被证明对一般对流扩散问题具有一阶收敛性。本文探讨利用SDC提升改进BUG方法的收敛阶数。在SDC框架中，我们首先通过改进BUG方法计算一阶解，随后通过求解皮卡积分方程的低秩解进行逐次修正。相较于直接应用SDC与改进BUG方法的简单组合，我们提出两个提升计算效率的改进方向：第一，通过详细分析截断参数对校正层级的依赖关系来降低中间数值秩；第二，在校正过程中精心选择子空间以避免求解大型线性系统（源自BUG方法的K步与L步）。我们证明所得格式对利普希茨连续有界动力系统具有高阶精度。通过比较两种低秩截断策略——基于截断奇异值分解的硬阈值策略与基于软阈值处理的软阈值策略，我们在框架中实现了数值秩控制。数值实验表明，对于弱耗散（或非耗散）问题的高阶格式，软阈值策略能提供更优的秩控制特性。