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Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order

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Abstract

Strong stability preserving (SSP) time discretizations preserve the monotonicity properties satisfied by the spatial discretization when coupled with the first order forward Euler, under a certain time-step restriction. The search for high order strong stability preserving time-stepping methods with high order and large allowable time-step has been an active area of research. It is known that implicit SSP Runge–Kutta methods exist only up to sixth order; however, if we restrict ourselves to solving only linear autonomous problems, the order conditions simplify and we can find implicit SSP Runge–Kutta methods of any linear order. In the current work we find implicit SSP Runge–Kutta methods with high linear order \(p_{lin} \le 9\) and nonlinear orders \(p=2,3,4\), that are optimal in terms of allowable SSP time-step. Next, we formulate a novel optimization problem for implicit–explicit (IMEX) SSP Runge–Kutta methods and find optimized IMEX SSP Runge–Kutta pairs that have high linear order \(p_{lin} \le 7\) and nonlinear orders up to \(p=4\). We also find implicit methods with large linear stability regions that pair with known explicit SSP Runge–Kutta methods. These methods are then tested on sample problems to demonstrate the sharpness of the SSP coefficient and the typical behavior of these methods on test problems.

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Acknowledgements

The authors wish to thank the anonymous referees whose careful reading and insightful comments dramatically improved this manuscript. This work was supported by AFOSR Grant FA9550-15-1-0235, and was partially supported by DOE NNSA ASC Algorithms effort, the DOE Office of Science AMR program at Sandia National Laboratory under contract DE-AC04-94AL85000.

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Authors and Affiliations

  1. Department of Mathematics, University of Massachusetts, North Dartmouth, MA, USA

    Sidafa Conde, Sigal Gottlieb & Zachary J. Grant

  2. Computational Mathematics Department, Sandia National Laboratories, Albuquerque, NM, USA

    Sidafa Conde & John N. Shadid

  3. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, USA

    John N. Shadid

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  1. Sidafa Conde
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  2. Sigal Gottlieb
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  3. Zachary J. Grant
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  4. John N. Shadid
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Correspondence to Sigal Gottlieb.

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Conde, S., Gottlieb, S., Grant, Z.J. et al. Implicit and Implicit–Explicit Strong Stability Preserving Runge–Kutta Methods with High Linear Order. J Sci Comput 73, 667–690 (2017). https://doi.org/10.1007/s10915-017-0560-2

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  • DOI: https://doi.org/10.1007/s10915-017-0560-2

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