An energy-preserving algorithm for nonlinear Hamiltonian wave equations with Neumann boundary conditions

Abstract

In this paper, a novel energy-preserving numerical scheme for nonlinear Hamiltonian wave equations with Neumann boundary conditions is proposed and analyzed based on the blend of spatial discretization by finite element method (FEM) and time discretization by Average Vector Field (AVF) approach. We first use the finite element discretization in space, which leads to a system of Hamiltonian ODEs whose Hamiltonian can be thought of as the semi-discrete energy of the original continuous system. The stability of the semi-discrete finite element scheme is analyzed. We then apply the AVF approach to the Hamiltonian ODEs to yield a new and efficient fully discrete scheme, which can preserve exactly (machine precision) the semi-discrete energy. The blend of FEM and AVF approach derives a new and efficient numerical scheme for nonlinear Hamiltonian wave equations. The numerical results on a single-soliton problem and a sine-Gordon equation are presented to demonstrate the remarkable energy-preserving property of the proposed numerical scheme.