On the efficiency of spectral deferred correction methods for time-dependent partial differential equations

Many physical and biological systems involve the interactions of two or more processes with widely-differing characteristic time scales. Previously, high-order semi-implicit and multi-implicit formulations of the spectral deferred correction methods (denoted by SISDC and MISDC methods, respectively) have been proposed for solving partial differential equations arising in such model systems. These methods compute a temporally high-order approximation by means of a first-order numerical method, which solves a series of correction equations to increase the temporal order Of accuracy of the approximation. MISDC methods also allow several fast-evolving processes to be handled implicitly but independently, allowing for different time steps for each process while avoiding the splitting errors present in traditional operator-splitting methods. In this study, we propose MISDC methods that use second- and third-order integration and splitting methods in the prediction steps, and we assess the efficiency of SISDC and MISDC methods that are based on those moderate-order integration methods. Numerical results indicate that SISDC methods using third-order prediction steps are the most efficient, but the efficiency of SISDC methods using first-order steps improves, particularly in higher spatial dimensions, when combined with a "ladder approach" that uses a less refined spatial discretization during the initial SDC iterations. Among the MISDC methods studied, the one with a third-order prediction step is the most efficient for a mildly-stiff problem, but the method with a first-order prediction step has the least splitting error and thus the highest efficiency for a stiff problem. Furthermore, a MISDC method using a second-order prediction step with Strang splitting generates approximations with large splitting errors, compared with methods that use a different operator-splitting approach that orders the integration of processes according to their relative stiffness. (C) 2008 IMACS. Published by Elsevier B.V. All rights reserved.