A discontinuous Galerkin method for the Camassa-Holm-Kadomtsev-Petviashvili type equations

This paper develops a high-order discontinuous Galerkin (DG) method for the Camassa-Holm-Kadomtsev-Petviashvili (CH-KP) type equations on Cartesian meshes. The significant part of the simulation for the CH-KP type equations lies in the treatment for the integration operator partial differential -1$$ {\partial}<^>{-1} $$. Our proposed DG method deals with it element by element, which is efficient and applicable to most solutions. Using the instinctive energy of the original PDE as a guiding principle, the DG scheme can be proved as an energy stable numerical scheme. In addition, the semi-discrete error estimates results for the nonlinear case are derived without any priori assumption. Several numerical experiments demonstrate the capability of our schemes for various types of solutions.