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On a family of discontinuous Galerkin fully-discrete schemes for the wave equation

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Abstract

In this paper, we study a family of discontinuous Galerkin (DG) fully discrete schemes for solving the second-order wave equation. The spatial variable discretization is based on an application of the DG method. The temporal variable discretization depends on a parameter \(\theta \in [0,1]\). Under suitable regularity hypotheses on the solution, optimal order error bounds are shown for the numerical schemes with \(\theta \in [\frac{1}{2},1]\), unconditionally with respect to the spatial mesh-size and the time-step, and for the numerical schemes with \(\theta \in [0,\frac{1}{2})\) where a Courant–Friedrichs–Lewy stability condition is satisfied relating the mesh-size and the time-step. The optimal order error estimates are derived for \(H^{1}(\Omega )\) and \(L^{2}(\Omega )\) norms. Simulation results are reported to provide numerical evidence of the optimal convergence orders predicted by the theory.

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Authors and Affiliations

  1. School of Science, Inner Mongolia University of Science and Technology, Baotou, 014010, Inner Mongolia, China

    Limin He

  2. School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, 710049, Shaanxi, China

    Limin He, Weimin Han & Fei Wang

  3. Department of Mathematics and Program in Applied Mathematical and Computational Sciences, University of Iowa, Iowa City, IA, 52242, USA

    Weimin Han

Authors
  1. Limin He
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  2. Weimin Han
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  3. Fei Wang
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Correspondence to Fei Wang.

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Communicated by Pierangelo Marcati.

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Limin He: The work of this author was partially supported by the National Natural Science Foundation of China (Grant No. 11801287, 62065015).

Fei Wang: The work of this author was partially supported by the National Natural Science Foundation of China (Grant No. 11771350).

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He, L., Han, W. & Wang, F. On a family of discontinuous Galerkin fully-discrete schemes for the wave equation. Comp. Appl. Math. 40, 56 (2021). https://doi.org/10.1007/s40314-021-01423-8

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  • DOI: https://doi.org/10.1007/s40314-021-01423-8

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