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In this paper, an energy-conservation scheme is derived and analysed for solving Hamiltonian wave equations subject to Neumann boundary conditions in two dimensions. The energy-conservation scheme is based on the blend of spatial discretisation by a fourth-order finite difference method and time integration by the Average Vector Field (AVF) approach. The spatial discretisation via the fourth-order finite difference leads to a particular Hamiltonian system of second-order ordinary differential equations. The conservative law of the discrete energy is established, and the stability and convergence of the semi-discrete scheme are analysed. For the time discretisation, the corresponding AVF formula is derived and applied to the particular Hamiltonian ODEs to yield an efficient energy-conservation scheme. The numerical simulation is implemented for various cases including a linear wave equation and two nonlinear sine-Gordon equations. The numerical results demonstrate the spatial accuracy and the remarkable energy-conservation behaviour of the proposed energy-conservation scheme in this paper.
}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/12806.html} }In this paper, an energy-conservation scheme is derived and analysed for solving Hamiltonian wave equations subject to Neumann boundary conditions in two dimensions. The energy-conservation scheme is based on the blend of spatial discretisation by a fourth-order finite difference method and time integration by the Average Vector Field (AVF) approach. The spatial discretisation via the fourth-order finite difference leads to a particular Hamiltonian system of second-order ordinary differential equations. The conservative law of the discrete energy is established, and the stability and convergence of the semi-discrete scheme are analysed. For the time discretisation, the corresponding AVF formula is derived and applied to the particular Hamiltonian ODEs to yield an efficient energy-conservation scheme. The numerical simulation is implemented for various cases including a linear wave equation and two nonlinear sine-Gordon equations. The numerical results demonstrate the spatial accuracy and the remarkable energy-conservation behaviour of the proposed energy-conservation scheme in this paper.