A direct discontinuous Galerkin method for the generalized Korteweg–de Vries equation: Energy conservation and boundary effect

A direct discontinuous Galerkin method for solving the generalized Korteweg–de Vries (KdV) equation with both periodic data and non-homogeneous boundary data is developed. A class of numerical fluxes is identified so that the method preserves both momentum and energy for the initial value problem. The method with such a property is numerically shown robust with less error in both phase and amplitude after long time simulation. Numerical examples are given to confirm the theoretical result and the capacity of this method for capturing soliton wave phenomena and various boundary wave patterns.