High-Order Implicit Low-Rank Method with Spectral Deferred Correction for Matrix Differential Equations

Abstract: 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.