New energy-preserving algorithms for nonlinear Hamiltonian wave equation equipped with Neumann boundary conditions

In this paper, using the blend of spatial discretization by second-order or fourth-order finite difference methods (FDM) and time integration by the generalized Average Vector Field (GAVF) method or the generalized adapted Average Vector Field (GAAVF) method, we propose and analyze novel energy-preserving algorithms for solving the nonlinear Hamiltonian wave equation equipped with homogeneous Neumann boundary conditions. Firstly, two kinds of finite difference methods are considered to discretize the spatial derivative, which can be of order two and order four respectively in all the spatial grid points. The conservation laws of the discrete energy are established after the semi-discretization, a Hamiltonian system of ODEs is derived whose Hamiltonian can be regarded as the approximate energy of the original continuous system. Then, the GAVF formula and the GAAVF formula are developed and applied to the derived Hamiltonian ODEs to yield some novel and efficient algorithms, which can exactly preserve the discrete energy. The numerical simulation is implemented and the numerical results demonstrate the spatial and temporal accuracy and the remarkable energy-preserving property of the new algorithms presented in this paper.