Vol. 5, No. 2, 2010

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A hybrid parareal spectral deferred corrections method

Michael L. Minion

Vol. 5 (2010), No. 2, 265–301

The parareal algorithm introduced in 2001 by Lions, Maday, and Turinici is an iterative method for the parallelization of the numerical solution of ordinary differential equations or partial differential equations discretized in the temporal direction. The temporal interval of interest is partitioned into successive domains which are assigned to separate processor units. Each iteration of the parareal algorithm consists of a high accuracy solution procedure performed in parallel on each domain using approximate initial conditions and a serial step which propagates a correction to the initial conditions through the entire time interval. The original method is designed to use classical single-step numerical methods for both of these steps. This paper investigates a variant of the parareal algorithm first outlined by Minion and Williams in 2008 that utilizes a deferred correction strategy within the parareal iterations. Here, the connections between parareal, parallel deferred corrections, and a hybrid parareal-spectral deferred correction method are further explored. The parallel speedup and efficiency of the hybrid methods are analyzed, and numerical results for ODEs and discretized PDEs are presented to demonstrate the performance of the hybrid approach.

parallel in time, parareal, ordinary differential equations, parallel computing, spectral deferred corrections
Mathematical Subject Classification 2000
Primary: 65L99
Received: 18 January 2010
Revised: 16 October 2010
Accepted: 19 October 2010
Published: 5 January 2011
Michael L. Minion
Department of Mathematics
University of North Carolina – Chapel Hill
Campus Box 3250
Chapel Hill, NC 27514-3250
 United States