Fourier Spectral Methods for Degasperis-Procesi Equation with Discontinuous Solutions

In this paper, we develop, analyze and test the Fourier spectral methods for solving the Degasperis–Procesi (DP) equation which contains nonlinear high order derivatives, and possibly discontinuous or sharp transition solutions. The L2 stability is obtained for general numerical solutions of the Fourier Galerkin method and Fourier collocation (pseudospectral) method. By applying the Gegenbauer reconstruction technique as a post-processing method to the Fourier spectral solution, we reduce the oscillations arising from the discontinuity successfully. The numerical simulation results for different types of solutions of the nonlinear DP equation are provided to illustrate the accuracy and capability of the methods.