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Explicit strong stability preserving multistep Runge-Kutta methods


Authors: Christopher Bresten, Sigal Gottlieb, Zachary Grant, Daniel Higgs, David I. Ketcheson and Adrian Németh
Journal: Math. Comp. 86 (2017), 747-769
MSC (2010): Primary 65M20
DOI: https://doi.org/10.1090/mcom/3115
Published electronically: June 2, 2016
MathSciNet review: 3584547
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Abstract: High-order spatial discretizations of hyperbolic PDEs are often designed to have strong stability properties, such as monotonicity. We study explicit multistep Runge-Kutta strong stability preserving (SSP) time integration methods for use with such discretizations. We prove an upper bound on the SSP coefficient of explicit multistep Runge-Kutta methods of order two and above. Numerical optimization is used to find optimized explicit methods of up to five steps, eight stages, and tenth order. These methods are tested on the linear advection and nonlinear Buckley-Leverett equations, and the results for the observed total variation diminishing and/or positivity preserving time-step are presented.


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Additional Information

Christopher Bresten
Affiliation: Department of Mathematics, University of Massachusetts, Dartmouth, 285 Old Westport Road, North Dartmouth, Massachusetts 02747
Email: cbresten@umassd.edu

Sigal Gottlieb
Affiliation: Department of Mathematics, University of Massachusetts, Dartmouth, 285 Old Westport Road, North Dartmouth, Massachusetts 02747
Email: sgottlieb@umassd.edu

Zachary Grant
Affiliation: Department of Mathematics, University of Massachusetts, Dartmouth, 285 Old Westport Road, North Dartmouth Massachusetts 02747
Email: zgrant@umassd.edu

Daniel Higgs
Affiliation: Department of Mathematics, University of Massachusetts, Dartmouth, 285 Old Westport Road, North Dartmouth, Massachusetts 02747

David I. Ketcheson
Affiliation: King Abdullah University of Science & Technology (KAUST), Thuwal, Saudi Arabia

Adrian Németh
Affiliation: Department of Mathematics and Computational Sciences, Széchenyi István University, Győr, Hungary

DOI: https://doi.org/10.1090/mcom/3115
Received by editor(s): September 3, 2014
Received by editor(s) in revised form: July 2, 2015, and September 18, 2015
Published electronically: June 2, 2016
Additional Notes: This research was supported by AFOSR grant number FA-9550-12-1-0224 and KAUST grant FIC/2010/05
Article copyright: © Copyright 2016 American Mathematical Society