High Order Energy Stable Local Discontinuous Galerkin Methods for Camassa–Holm–Novikov Equations

Abstract This paper presents and analyzes energy stable local discontinuous Galerkin (LDG) methods for solving the cubic nonlinear shallow water wave equations, also referred to as the Camassa–Holm–Novikov (CHN) equations. These methods are characterized by their high-order accuracy and effectiveness in addressing the intricate nonlinear terms inherent to the CHN equations. The success of the proposed numerical schemes is due to the careful selection of the numerical fluxes at the cell boundaries, which are crucial for ensuring stability and accuracy. We employ a specific interpolation function to derive optimal a priori error estimates and provide a comprehensive analysis of the quadratic Taylor expansion of the cubic nonlinearity. In addition, our numerical schemes show excellent performance for different peakon solutions. Numerical experiments validate the accuracy and efficiency of the proposed methods, demonstrating their effectiveness in practical applications.