An energy-conserving ultra-weak discontinuous Galerkin method for the generalized Korteweg–de Vries equation

Abstract

Abstract We propose an energy-conserving ultra-weak discontinuous Galerkin (DG) method for the generalized Korteweg– de Vries (KdV) equation in one dimension. Optimal a priori error estimate of order k + 1 is obtained for the semi-discrete scheme for the KdV equation without the convection term on general nonuniform meshes when polynomials of degree k ≥ 2 are used. We also numerically observe optimal convergence of the method for the KdV equation with linear or nonlinear convection terms. It is numerically observed for the new method to have a superior performance for long-time simulations over existing DG methods.