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Optimal Order Error Estimates for Discontinuous Galerkin Methods for the Wave Equation

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Abstract

In this paper, we derive optimal order error estimates for spatially semi-discrete and fully discrete schemes to numerically solve the second-order wave equation. The numerical schemes are constructed with the discontinuous Galerkin (DG) discretization for the spatial variable and the centered second-order finite difference approximation for the temporal variable. Under appropriate regularity assumptions on the solution, the schemes are shown to enjoy the optimal order error bounds in terms of both the spatial mesh-size and the time-step. In Grote and Schötzau (J Sci Comput 40:257–272, 2009), a fully discrete DG scheme is studied with an explicit finite difference temporal discretization where a CFL condition is required on the mesh-size and the time-step, and optimal order error estimates are derived in the \(L^2(\Omega )\)-norm. In comparison, for our fully discrete DG schemes, we do not require a CFL condition on the mesh-size and the time-step, and our optimal order error estimates are derived for the \(H^1(\Omega )\)-like norm and the \(L^2(\Omega )\) norm. Numerical simulation results are reported to illustrate theoretically predicted convergence orders in the \(H^1(\Omega )\) and \(L^2(\Omega )\) norms.

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Authors and Affiliations

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

    Weimin Han, Limin He & Fei Wang

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

    Weimin Han

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

    Limin He

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

Additional information

The work of Weimin Han was partially supported by NSF under Grant DMS-1521684. The work of Limin He, Fei Wang was partially supported by the National Natural Science Foundation of China (Grant Nos. 61663035, 11771350).

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Han, W., He, L. & Wang, F. Optimal Order Error Estimates for Discontinuous Galerkin Methods for the Wave Equation. J Sci Comput 78, 121–144 (2019). https://doi.org/10.1007/s10915-018-0755-1

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  • DOI: https://doi.org/10.1007/s10915-018-0755-1

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