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Discontinuous Galerkin Methods for the Ostrovsky–Vakhnenko Equation

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Abstract

In this paper, we develop discontinuous Galerkin methods for the Ostrovsky–Vakhnenko (OV) equation, which yields the shock solutions and singular soliton solutions, such as peakon, cuspon and loop solitons. The OV equation has also been shown to have a bi-Hamiltonian structure. We directly develop the energy stable or Hamiltonian conservative discontinuous Galerkin schemes for the OV equation. Error estimates for the two energy stable schemes are also proved. For some singular solutions, including cuspon and loop soliton solutions, the hodograph transformation is adopted to transform the OV equation or the generalized OV system to the coupled dispersionless (CD) system. Subsequently, two discontinuous Galerkin schemes are constructed for the transformed CD system. Numerical experiments are provided to demonstrate the accuracy and capability of the proposed schemes, including shock solution and, peakon, cuspon and loop soliton solutions.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Science and Technology of China, Hefei, 230026, Anhui, People’s Republic of China

    Qian Zhang & Yinhua Xia

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  1. Qian Zhang
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  2. Yinhua Xia
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Yinhua Xia: Research supported by NSFC Grant 11871449, and a Grant from Laboratory of Computational Physics (No. 6142A0502020817).

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Zhang, Q., Xia, Y. Discontinuous Galerkin Methods for the Ostrovsky–Vakhnenko Equation. J Sci Comput 82, 24 (2020). https://doi.org/10.1007/s10915-019-01109-8

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  • DOI: https://doi.org/10.1007/s10915-019-01109-8

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