Achieving algorithmic resilience for temporal integration through spectral deferred corrections
Abstract
Spectral deferred corrections (SDC) is an iterative approach for constructing higherorderaccurate 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 lowerorder 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 NavierStokes equations for combustion research.
 Authors:
 National Renewable Energy Lab. (NREL), Golden, CO (United States). Computational Science Center
 Sandia National Lab. (SNLCA), Livermore, CA (United States). Scalable Modeling and Analysis Dept.
 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States). Computational Research Division
 Publication Date:
 Research Org.:
 Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States); National Renewable Energy Lab. (NREL), Golden, CO (United States); Sandia National Lab. (SNLCA), Livermore, CA (United States)
 Sponsoring Org.:
 USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC21)
 OSTI Identifier:
 1371646
 Report Number(s):
 NREL/JA2C0062926; NREL/JA2C0068888
Journal ID: ISSN 15593940
 DOE Contract Number:
 AC0205CH11231; AC3608GO28308
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Communications in Applied Mathematics and Computational Science; Journal Volume: 12; Journal Issue: 1
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; SDC; deferred correction; resilience; time integration; combustion
Citation Formats
Grout, Ray, Kolla, Hemanth, Minion, Michael, and Bell, John. Achieving algorithmic resilience for temporal integration through spectral deferred corrections. United States: N. p., 2017.
Web. doi:10.2140/camcos.2017.12.25.
Grout, Ray, Kolla, Hemanth, Minion, Michael, & Bell, John. Achieving algorithmic resilience for temporal integration through spectral deferred corrections. United States. doi:10.2140/camcos.2017.12.25.
Grout, Ray, Kolla, Hemanth, Minion, Michael, and Bell, John. Mon .
"Achieving algorithmic resilience for temporal integration through spectral deferred corrections". United States.
doi:10.2140/camcos.2017.12.25.
@article{osti_1371646,
title = {Achieving algorithmic resilience for temporal integration through spectral deferred corrections},
author = {Grout, Ray and Kolla, Hemanth and Minion, Michael and Bell, John},
abstractNote = {Spectral deferred corrections (SDC) is an iterative approach for constructing higherorderaccurate 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 lowerorder 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 NavierStokes equations for combustion research.},
doi = {10.2140/camcos.2017.12.25},
journal = {Communications in Applied Mathematics and Computational Science},
number = 1,
volume = 12,
place = {United States},
year = {Mon May 08 00:00:00 EDT 2017},
month = {Mon May 08 00:00:00 EDT 2017}
}

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 lowerorder 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 tomore »

Achieving algorithmic resilience for temporal integration through spectral deferred corrections
Spectral deferred corrections (SDC) is an iterative approach for constructing higherorderaccurate 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 lowerorder 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 frommore » 
A Gas Dynamics Method Based on The Spectral Deferred Corrections (SDC) Time Integration Technique and The Piecewise Parabolic Method (PPM)
We present a computational gas dynamics method based on the Spectral Deferred Corrections (SDC) time integration technique and the Piecewise Parabolic Method (PPM) finite volume method. The PPM framework is used to define edge averaged quantities which are then used to evaluate numerical flux functions. The SDC technique is used to integrate solution in time. This kind of approach was first taken by Anita et al in [17]. However, [17] is problematic when it is implemented to certain shock problems. Here we propose significant improvements to [17]. The method is fourth order (both in space and time) for smooth flows,more » 
Semiimplicit spectral deferred correction methods for ordinary differential equations
A semiimplicit formulation of the method of spectral deferred corrections (SISDC) for ordinary differential equations with both stiff and nonstiff terms is presented. Several modifications and variations to the original spectral deferred corrections method by Dutt, Greengard, and Rokhlin concerning the choice of integration points and the form of the correction iteration are presented. The stability and accuracy of the resulting ODE methods are explored analytically and numerically. The SISDC methods are intended to be combined with the method of lines approach to yield a flexible framework for creating higherorder semiimplicit methods for partial differential equations. A discussion and numericalmore »