Modified Structure-Preserving Schemes for the Degasperis—Procesi Equation**Supported by the National Natural Science Foundation of China under Grant Nos 11571366 and 11501570, and the Open Foundation of State Key Laboratory of High Performance Computing of China.

Abstract

We investigate the structure-preserving numerical algorithm of the Degasperis—Procesi equation which can be transformed into a bi-Hamiltonian form using the discrete variational derivative method. Based on two different space discretization methods, the Fourier pseudospectral method and the wavelet collocation method, we develop two modified structure-preserving schemes under the periodic boundary condition. These proposed schemes are proved to preserve the Hamiltonian invariants theoretically and numerically. Meanwhile, the numerical results confirm that they can simulate the propagation of solitons effectively for a long time.