A Conservative Discontinuous Galerkin Method for the Degasperis–procesi Equation

In this work, we design, analyze and test a conservative discontinuous Galerkin method for solving the Degasperis–Procesi equation. This model is integrable and admits possibly discontinuous solutions, and therefore suitable for modeling both short wave breaking and long wave propagation phenomena. The proposed numerical method is high order accurate, and preserves two invariants, mass and energy, of this nonlinear equation, hence producing wave solutions with satisfying long time behavior. The L-stability of the scheme for general solutions is a consequence of the energy preserving property. The numerical simulation results for different types of solutions of the nonlinear Degasperis–Procesi equation are provided to illustrate the accuracy and capability of the method.