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Leapfrog Time-Stepping for Hermite Methods

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Abstract

We introduce Hermite-leapfrog methods for first order linear wave systems. The new Hermite-leapfrog methods pair leapfrog time-stepping with the Hermite methods of Goodrich and co-authors et al. (Math Comput 75(254):595–630, 2006). The new schemes stagger field variables in both time and space and are high-order accurate for equations with smooth solutions and coefficients. We provide a detailed description of the method and demonstrate that the method conserves variable quantities. Higher dimensional versions of the method are constructed via tensor products. Numerical evidence and rigorous analysis in one space dimension establish stability and high-order convergence. Experiments demonstrating efficient implementations on a graphics processing unit are also presented.

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Acknowledgements

The authors would like to thank Daniel Appelö for the fruitful conversations. TH is supported in part by NSF Grant DMS-1418871. Any conclusions or recommendations expressed in this paper are those of the author and do not necessarily reflect the views of the NSF. JC is supported by NSF Grants DMS-1719818 and DMS-1712639. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. LLNL-JRNL-757049.

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Authors and Affiliations

  1. Lawrence Livermore National Laboratory, 7000 East Ave, Livermore, CA, 94550, USA

    Arturo Vargas

  2. Department of Mathematics, Southern Methodist University, Dallas, TX, 75275, USA

    Thomas Hagstrom

  3. Department of Computational and Applied Mathematics, Rice University, 6100 Main Street, Houston, TX, 77005, USA

    Jesse Chan

  4. Department of Mathematics, Virginia Tech, Blacksburg, VA, 24060, USA

    Tim Warburton

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  1. Arturo Vargas
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  2. Thomas Hagstrom
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  4. Tim Warburton
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Correspondence to Arturo Vargas.

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Vargas, A., Hagstrom, T., Chan, J. et al. Leapfrog Time-Stepping for Hermite Methods. J Sci Comput 80, 289–314 (2019). https://doi.org/10.1007/s10915-019-00938-x

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