Energy-Preserving Algorithms for the Benjamin Equation

This paper presents several hybrid algorithms to preserve the global energy of the Benjamin equation. The Benjamin equation is a non-local partial differential equation involving the Hilbert transform. For this sake, quite few structure-preserving integrators have been proposed so far. Our schemes are derived based on an extended multi-symplectic Hamiltonian system of the Benjamin equation by using Fourier pseudospectral method, finite element method and wavelet collocation method spatially coupled with the AVF method temporally. The local and global properties of the proposed schemes are studied. Numerical experiments are presented to demonstrate the conservative properties of the proposed numerical methods and study the evolutions of the numerical solutions of solitary waves and wave breaking.