Downwinding for preserving strong stability in explicit integrating factor Runge-Kutta methods
Abstract
Here strong stability preserving (SSP) Runge-Kutta methods are desirable when evolving in time problems that have discontinuities or sharp gradients and require nonlinear non-inner-product stability properties to be satisfied. Unlike the case for L2 linear stability, implicit methods do not significantly alleviate the time-step restriction when the SSP property is needed. For this reason, when handling problems with a linear component that is stiff and a nonlinear component that is not, SSP integrating factor Runge-Kutta methods may offer an attractive alternative to traditional time-stepping methods. The strong stability properties of integrating factor Runge-Kutta methods where the transformed problem is evolved with an explicit SSP Runge-Kutta method with non-decreasing abscissas was recently established. However, these methods typically have smaller SSP coefficients (and therefore a smaller allowable time-step) than the optimal SSP Runge-Kutta methods, which often have some decreasing abscissas. In this work, we consider the use of downwinded spatial operators to preserve the strong stability properties of integrating factor Runge-Kutta methods where the Runge-Kutta method has some decreasing abscissas. We present the SSP theory for this approach and present numerical evidence to show that such an approach is feasible and performs as expected. However, we also show that in some casesmore »
- Authors:
-
[2];
- Univ. of Massachusetts, Dartmouth, MA (United States)
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- Publication Date:
- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); Air Force Office of Scientific and Research (AFOSR); National Science Foundation (NSF)
- OSTI Identifier:
- 1844914
- Grant/Contract Number:
- AC05-00OR22725; FA9550-15-1-0235: DMS-1719698
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Pure and Applied Mathematics Quarterly
- Additional Journal Information:
- Journal Volume: 14; Journal Issue: 1; Journal ID: ISSN 1558-8599
- Publisher:
- International Press
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 97 MATHEMATICS AND COMPUTING
Citation Formats
Isherwood, Leah, Grant, Zachary J., and Gottlieb, Sigal. Downwinding for preserving strong stability in explicit integrating factor Runge-Kutta methods. United States: N. p., 2019.
Web. doi:10.4310/pamq.2018.v14.n1.a1.
Isherwood, Leah, Grant, Zachary J., & Gottlieb, Sigal. Downwinding for preserving strong stability in explicit integrating factor Runge-Kutta methods. United States. https://doi.org/10.4310/pamq.2018.v14.n1.a1
Isherwood, Leah, Grant, Zachary J., and Gottlieb, Sigal. Tue .
"Downwinding for preserving strong stability in explicit integrating factor Runge-Kutta methods". United States. https://doi.org/10.4310/pamq.2018.v14.n1.a1. https://www.osti.gov/servlets/purl/1844914.
@article{osti_1844914,
title = {Downwinding for preserving strong stability in explicit integrating factor Runge-Kutta methods},
author = {Isherwood, Leah and Grant, Zachary J. and Gottlieb, Sigal},
abstractNote = {Here strong stability preserving (SSP) Runge-Kutta methods are desirable when evolving in time problems that have discontinuities or sharp gradients and require nonlinear non-inner-product stability properties to be satisfied. Unlike the case for L2 linear stability, implicit methods do not significantly alleviate the time-step restriction when the SSP property is needed. For this reason, when handling problems with a linear component that is stiff and a nonlinear component that is not, SSP integrating factor Runge-Kutta methods may offer an attractive alternative to traditional time-stepping methods. The strong stability properties of integrating factor Runge-Kutta methods where the transformed problem is evolved with an explicit SSP Runge-Kutta method with non-decreasing abscissas was recently established. However, these methods typically have smaller SSP coefficients (and therefore a smaller allowable time-step) than the optimal SSP Runge-Kutta methods, which often have some decreasing abscissas. In this work, we consider the use of downwinded spatial operators to preserve the strong stability properties of integrating factor Runge-Kutta methods where the Runge-Kutta method has some decreasing abscissas. We present the SSP theory for this approach and present numerical evidence to show that such an approach is feasible and performs as expected. However, we also show that in some cases the integrating factor approach with explicit SSP Runge-Kutta methods with non-decreasing abscissas performs nearly as well, if not better, than with explicit SSP Runge-Kutta methods with downwinding. In conclusion, while the downwinding approach can be rigorously shown to guarantee the SSP property for a larger timestep, in practice using the integrating factor approach by including downwinding as needed with optimal explicit SSP Runge-Kutta methods does not necessarily provide significant benefit over using explicit SSP Runge-Kutta methods with non-decreasing abscissas.},
doi = {10.4310/pamq.2018.v14.n1.a1},
journal = {Pure and Applied Mathematics Quarterly},
number = 1,
volume = 14,
place = {United States},
year = {Tue Apr 02 00:00:00 EDT 2019},
month = {Tue Apr 02 00:00:00 EDT 2019}
}
Search WorldCat to find libraries that may hold this journalWorks referencing / citing this record:
Strong Stability Preserving Integrating Factor Two-Step Runge–Kutta Methods
journal, September 2019
- Isherwood, Leah; Grant, Zachary J.; Gottlieb, Sigal
- Journal of Scientific Computing, Vol. 81, Issue 3