Error estimates of the local discontinuous Galerkin methods for two-dimensional ([formula omitted])-Camassa–Holm equations

Abstract

In this paper, we present the uniform framework of local discontinuous Galerkin (LDG) methods for two-dimensional Camassa–Holm equations and two-dimensional μ-Camassa–Holm equations. The energy stability and the semi-discrete error estimates based on the uniform framework for two equations are derived. The optimal error estimates with order k for approximating the first-order derivatives with Qk elements in Cartesian meshes are obtained. Compared with the error estimates for one-dimensional cases, more auxiliary variables and inter-element jump terms make the derivation more complicated. Numerical experiments for different circumstances are displayed to illustrate the accuracy and stability of those schemes.