The Conservative Time High-Order AVF Compact Finite Difference Schemes for Two-Dimensional Variable Coefficient Acoustic Wave Equations

Abstract

In this paper, we develop and analyze the energy conservative time high-order AVF compact finite difference methods for variable coefficient acoustic wave equations in two dimensions. We first derive out an infinite-dimensional Hamiltonian system for the variable coefficient wave equations and apply the spatial fourth-order compact finite difference operator to the equations of the system to obtain a semi-discrete approximation system, which can be cast into a canonical finite-dimensional Hamiltonian form. We then apply the second-order and fourth-order AVF techniques to propose the fully discrete energy conservative time high-order AVF compact finite difference methods for wave equations in two dimensions. We prove that the proposed semi-discrete and fully-discrete schemes satisfy energy conservations in the discrete forms. We further prove that the semi-discrete scheme has the fourth-order convergence order in space and the fully-discrete AVF compact finite difference method has the fourth-order convergence order in both time and space. Numerical tests confirm the theoretical results.