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

Abstract

AbstractThis 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 . 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.