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Local Discontinuous Galerkin Methods for the Two-Dimensional Camassa–Holm Equation

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Abstract

In this paper, the local discontinuous Galerkin method is developed to solve the two-dimensional Camassa–Holm equation in rectangular meshes. The idea of LDG methods is to suitably rewrite a higher-order partial differential equations into a first-order system, then apply the discontinuous Galerkin method to the system. A key ingredient for the success of such methods is the correct design of interface numerical fluxes. The energy stability for general solutions of the method is proved. Comparing with the Camassa–Holm equation in one-dimensional case, there are more auxiliary variables which are introduced to handle high-order derivative terms. The proof of the stability is more complicated. The resulting scheme is high-order accuracy and flexible for arbitrary h and p adaptivity. Different types of numerical simulations are provided to illustrate the accuracy and stability of the method.

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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

    Tian Ma & Yan Xu

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  1. Tian Ma
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  2. Yan Xu
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Correspondence to Yan Xu.

Additional information

Research supported by NSFC Grant Nos. 11722112, 91630207.

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Ma, T., Xu, Y. Local Discontinuous Galerkin Methods for the Two-Dimensional Camassa–Holm Equation. Commun. Math. Stat. 6, 359–388 (2018). https://doi.org/10.1007/s40304-018-0140-2

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  • DOI: https://doi.org/10.1007/s40304-018-0140-2

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