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Strong Stability Preserving Explicit Peer Methods for Discontinuous Galerkin Discretizations

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Abstract

In this paper we study explicit peer methods with the strong stability preserving (SSP) property for the numerical solution of hyperbolic conservation laws in one space dimension. A system of ordinary differential equations is obtained by discontinuous Galerkin (DG) spatial discretizations, which are often used in the method of lines approach to solve hyperbolic differential equations. We present in this work the construction of explicit peer methods with stability regions that are designed for DG spatial discretizations and with large SSP coefficients. Methods of second- and third order with up to six stages are optimized with respect to both properties. The methods constructed are tested and compared with appropriate Runge–Kutta methods. The advantage of high stage order is verified numerically.

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Acknowledgements

The authors are grateful to the anonymous referees for their valuable remarks and comments on the paper.

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Authors and Affiliations

  1. Institute of Mathematics, Martin Luther University Halle-Wittenberg, D-06099, Halle (Saale), Germany

    Marcel Klinge & Rüdiger Weiner

Authors
  1. Marcel Klinge
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  2. Rüdiger Weiner
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Correspondence to Marcel Klinge.

Appendix: Coefficients of the New DGSSP-Optimized Explicit Peer Methods

Appendix: Coefficients of the New DGSSP-Optimized Explicit Peer Methods

See Tables 5, 6, 7, 8, 9, 10, 11, 12.

Table 5 DGSSP-optimized explicit peer method, \(s=2,\ p=2\) with DG(2) spatial operator
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Table 6 DGSSP-optimized explicit peer method, \(s=3,\ p=2\) with DG(2) spatial operator
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Table 7 DGSSP-optimized explicit peer method, \(s=4,\ p=2\) with DG(2) spatial operator
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Table 8 DGSSP-optimized explicit peer method, \(s=5,\ p=2\) with DG(2) spatial operator
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Table 9 DGSSP-optimized explicit peer method, \(s=6,\ p=2\) with DG(2) spatial operator
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Table 10 DGSSP-optimized explicit peer method, \(s=3,\ p=3\) with DG(3) spatial operator
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Table 11 DGSSP-optimized explicit peer method, \(s=4,\ p=3\) with DG(3) spatial operator
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Table 12 DGSSP-optimized explicit peer method, \(s=5,\ p=3\) with DG(3) spatial operator
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Klinge, M., Weiner, R. Strong Stability Preserving Explicit Peer Methods for Discontinuous Galerkin Discretizations. J Sci Comput 75, 1057–1078 (2018). https://doi.org/10.1007/s10915-017-0573-x

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  • DOI: https://doi.org/10.1007/s10915-017-0573-x

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