A new high-order energy-preserving scheme for the modified Korteweg-de Vries equation

In this paper, a new high-order energy-preserving scheme is proposed for the modified Korteweg-de Vries equation. The proposed scheme is constructed by using the Hamiltonian boundary value methods in time, and Fourier pseudospectral method in space. Exploiting this method, we get second-order and fourth-order energy-preserving integrators. The proposed schemes not only have high accuracy, but also exactly conserve the total mass and energy in the discrete level. Finally, single solitary wave and the interaction of two solitary waves are presented to illustrate the effectiveness of the proposed schemes.