Vol. 12, No. 1, 2017

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Achieving algorithmic resilience for temporal integration through spectral deferred corrections

Ray W. Grout, Hemanth Kolla, Michael L. Minion and John B. Bell

Vol. 12 (2017), No. 1, 25–50

Spectral deferred corrections (SDC) is an iterative approach for constructing higher-order-accurate numerical approximations of ordinary differential equations. SDC starts with an initial approximation of the solution defined at a set of Gaussian or spectral collocation nodes over a time interval and uses an iterative application of lower-order time discretizations applied to a correction equation to improve the solution at these nodes. Each deferred correction sweep increases the formal order of accuracy of the method up to the limit inherent in the accuracy defined by the collocation points. In this paper, we demonstrate that SDC is well suited to recovering from soft (transient) hardware faults in the data. A strategy where extra correction iterations are used to recover from soft errors and provide algorithmic resilience is proposed. Specifically, in this approach the iteration is continued until the residual (a measure of the error in the approximation) is small relative to the residual of the first correction iteration and changes slowly between successive iterations. We demonstrate the effectiveness of this strategy for both canonical test problems and a comprehensive situation involving a mature scientific application code that solves the reacting Navier–Stokes equations for combustion research.

SDC, deferred correction, resilience, time integration, combustion
Mathematical Subject Classification 2010
Primary: 65D30, 65M12, 65M22, 80A25, 94B99
Secondary: 65M20
Received: 3 March 2016
Revised: 9 September 2016
Accepted: 18 January 2017
Published: 8 May 2017
Ray W. Grout
Computational Science Center
National Renewable Energy Laboratory
15013 Denver West Parkway
Golden, CO 80401
United States
Hemanth Kolla
Sandia National Laboratories
P.O. Box 969, M.S. 9158
7011 East Ave
Livermore, CA 94551-0969
United States
Michael L. Minion
Center for Computational Sciences and Engineering
Lawrence Berkeley National Laboratory
MS 50A-3141
1 Cyclotron Road
Berkeley, CA 94720
United States
John B. Bell
Center for Computational Sciences and Engineering
Lawrence Berkeley National Laboratory
MS 50A-3141
1 Cyclotron Road
Berkeley, CA 94720
United States