# Achieving algorithmic resilience for temporal integration through spectral deferred corrections

## Abstract

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.

- Authors:

- National Renewable Energy Lab. (NREL), Golden, CO (United States). Computational Science Center
- Sandia National Lab. (SNL-CA), 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. (SNL-CA), Livermore, CA (United States)

- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR) (SC-21)

- OSTI Identifier:
- 1371646

- Report Number(s):
- NREL/JA-2C00-62926; NREL/JA-2C00-68888

Journal ID: ISSN 1559-3940

- DOE Contract Number:
- AC02-05CH11231; AC36-08GO28308

- 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 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.},

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}

}