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Optimal Strong-Stability-Preserving Runge–Kutta Time Discretizations for Discontinuous Galerkin Methods

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Abstract

Discontinuous Galerkin (DG) spatial discretizations are often used in a method-of-lines approach with explicit strong-stability-preserving (SSP) Runge–Kutta (RK) time steppers for the numerical solution of hyperbolic conservation laws. The time steps that are employed in this type of approach must satisfy Courant–Friedrichs–Lewy stability constraints that are dependent on both the region of absolute stability and the SSP coefficient of the RK method. While existing SSPRK methods have been optimized with respect to the latter, it is in fact the former that gives rise to stricter constraints on the time step in the case of RKDG stability. Therefore, in this work, we present the development of new “DG-optimized” SSPRK methods with stability regions that have been specifically designed to maximize the stable time step size for RKDG methods of a given order in one space dimension. These new methods represent the best available RKDG methods in terms of computational efficiency, with significant improvements over methods using existing SSPRK time steppers that have been optimized with respect to SSP coefficients. Second-, third-, and fourth-order methods with up to eight stages are presented, and their stability properties are verified through application to numerical test cases.

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Acknowledgments

The first and second author acknowledge support by National Science Foundation Grants DMS-0915118 and DMS-1217218.

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Authors and Affiliations

  1. Department of Civil, Environmental, and Geodetic Engineering, The Ohio State University, 2070 Neil Avenue, Columbus, OH, 43210, USA

    Ethan J. Kubatko & Benjamin A. Yeager

  2. Division of Mathematical and Computer Sciences and Engineering, 4700 King Abdullah University of Science and Technology, Thuwal, 23955, Saudi Arabia

    David I. Ketcheson

Authors
  1. Ethan J. Kubatko
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  2. Benjamin A. Yeager
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  3. David I. Ketcheson
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Corresponding author

Correspondence to Ethan J. Kubatko.

Appendix: Shu–Osher Form Coefficients of the New Runge–Kutta Methods

Appendix: Shu–Osher Form Coefficients of the New Runge–Kutta Methods

See Tables 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 and 21.

Table 7 SSPRK(3,2)
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Table 8 SSPRK(4,2)
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Table 9 SSPRK(5,2)
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Table 10 SSPRK(6,2)
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Table 11 SSPRK(7,2)
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Table 12 SSPRK(8,2)
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Table 13 SSPRK(4,3)
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Table 14 SSPRK(5,3)
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Table 15 SSPRK(6,3)
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Table 16 SSPRK(7,3)
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Table 17 SSPRK(8,3)
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Table 18 SSPRK(5,4)
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Table 19 SSPRK(6,4)
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Table 20 SSPRK(7,4)
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Table 21 SSPRK(8,4)
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Kubatko, E.J., Yeager, B.A. & Ketcheson, D.I. Optimal Strong-Stability-Preserving Runge–Kutta Time Discretizations for Discontinuous Galerkin Methods. J Sci Comput 60, 313–344 (2014). https://doi.org/10.1007/s10915-013-9796-7

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  • DOI: https://doi.org/10.1007/s10915-013-9796-7

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