Energy Stability of Explicit Runge--Kutta Methods for Nonautonomous or Nonlinear Problems

Type
Article

Authors
Ranocha, Hendrik
Ketcheson, David I.

KAUST Department
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Applied Mathematics and Computational Science Program

Preprint Posting Date
2019-09-29

Online Publication Date
2020-11-24

Print Publication Date
2020-01

Date
2020-11-24

Submitted Date
2019-09-30

Abstract
Many important initial value problems have the property that energy is nonincreasing in time. Energy stable methods, also referred to as strongly stable methods, guarantee the same property discretely. We investigate requirements for conditional energy stability of explicit Runge--Kutta methods for nonlinear or nonautonomous problems. We provide both necessary and sufficient conditions for energy stability over these classes of problems. Examples of conditionally energy stable schemes are constructed, and an example is given in which unconditional energy stability is obtained with an explicit scheme.

Citation
Ranocha, H., & Ketcheson, D. I. (2020). Energy Stability of Explicit Runge--Kutta Methods for Nonautonomous or Nonlinear Problems. SIAM Journal on Numerical Analysis, 58(6), 3382–3405. doi:10.1137/19m1290346

Acknowledgements
Research reported in this publication was supported by the King Abdullah University of Scienceand Technology (KAUST). The first author was partially supported by the German Research Foundation (DFG, Deutsche Forschungsgemeinschaft) under Grant SO 363/14-1.

Publisher
Society for Industrial & Applied Mathematics (SIAM)

Journal
SIAM Journal on Numerical Analysis

DOI
10.1137/19m1290346

arXiv
1909.13215

Additional Links
https://epubs.siam.org/doi/10.1137/19M1290346

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