TY - JOUR
T1 - Numerical Analysis of an Energy-Conservation Scheme for Two-Dimensional Hamiltonian Wave Equations with Neumann Boundary Conditions
AU - Liu , Changying
AU - Shi , Wei
AU - Wu , Xinyuan
JO - International Journal of Numerical Analysis and Modeling
VL - 2
SP - 319
EP - 339
PY - 2018
DA - 2018/10
SN - 16
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/12806.html
KW - Two-dimensional Hamiltonian wave equation, finite difference method, Neumann boundary conditions, energy-conservation algorithm, average vector field formula.
AB - <p style="text-align: justify;">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.</p>