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Optimal explicit strong stability preserving Runge-Kutta methods with high linear order and optimal nonlinear order


Authors: Sigal Gottlieb, Zachary Grant and Daniel Higgs
Journal: Math. Comp. 84 (2015), 2743-2761
MSC (2010): Primary 65-XX
DOI: https://doi.org/10.1090/mcom/2966
Published electronically: April 10, 2015
MathSciNet review: 3378846
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Abstract: High order spatial discretizations with monotonicity properties are often desirable for the solution of hyperbolic PDEs. These methods can advantageously be coupled with high order strong stability preserving time discretizations. The search for high order strong stability time-stepping methods with large allowable strong stability coefficient has been an active area of research over the last two decades. This research has shown that explicit SSP Runge-Kutta methods exist only up to fourth order. However, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and this order barrier is lifted: explicit SSP Runge-Kutta methods of any linear order exist. These methods reduce to second order when applied to nonlinear problems. In the current work we aim to find explicit SSP Runge-Kutta methods with large allowable time-step, that feature high linear order and simultaneously have the optimal fourth order nonlinear order. These methods have strong stability coefficients that approach those of the linear methods as the number of stages and the linear order is increased. This work shows that when a high linear order method is desired, it may still be worthwhile to use methods with higher nonlinear order.


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Additional Information

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

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

Daniel Higgs
Affiliation: Mathematics Department, University of Massachusetts Dartmouth, 285 Old Westport Road, North Dartmouth, Massachusetts 02747
Email: danielhiggs@gmail.com

DOI: https://doi.org/10.1090/mcom/2966
Received by editor(s): March 14, 2014
Published electronically: April 10, 2015
Article copyright: © Copyright 2015 American Mathematical Society