Local Discontinuous Galerkin Methods for the Degasperis-Procesi Equation

Abstract

In this paper, we develop, analyze and test local discontinuous Galerkin (LDG) methods for solving the Degasperis-Procesi equation which contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The LDG method has the flexibility for arbitrary h and p adaptivity. We prove the L² stability for general solutions. The proof of the total variation stability of the schemes for the piecewise constant P⁰ case is also given. 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 LDG method.