Error estimates of the local discontinuous Galerkin methods for two-dimensional (μ)-Camassa-Holm equations

In this paper, we present the uniform framework of local discontinuous Galerkin (LDG) methods for two-dimensional Camassa-Holm equations and two-dimensional mu-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. (c) 2022 Elsevier B.V. All rights reserved.