Vol. 2, No. 1, 2007

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Implications of the choice of predictors for semi-implicit Picard integral deferred correction methods

Anita T. Layton and Michael L. Minion

Vol. 2 (2007), No. 1, 1–34

High-order semi-implicit Picard integral deferred correction (SIPIDC) methods have previously been proposed for the time-integration of partial differential equations with two or more disparate time scales. The SIPIDC methods studied to date compute a high-order approximation by first computing a provisional solution with a first-order semi-implicit method and then using a similar semi-implicit method to solve a series of correction equations, each of which raises the order of accuracy of the solution by one. This study assesses the efficiency of SIPIDC methods that instead use standard semi-implicit methods with orders two through four to compute the provisional solution. Numerical results indicate that using a method with more than first-order accuracy in the computation of the provisional solution increases the efficiency of SIPIDC methods in some cases. First-order PIDC corrections can improve the efficiency of semi-implicit integration methods based on backward difference formulae (BDF) or Runge–Kutta methods while maintaining desirable stability properties. Finally, the phenomenon of order reduction, which may be encountered in the integration of stiff problems, can be partially alleviated by the use of BDF methods in the computation of the provisional solution.

semi-implicit methods, deferred correction methods, order reduction
Mathematical Subject Classification 2000
Primary: 65B05
Secondary: 65L20
Received: 5 December 2005
Accepted: 20 October 2006
Published: 2 August 2007
Anita T. Layton
Department of Mathematics
Duke University
Box 90320
Durham, NC 27708
United States
Michael L. Minion
Department of Mathematics
CB 3250 Phillips Hall
University of North Carolina
Chapel Hill, NC 27599
United States