Energy-preserving time high-order AVF compact finite difference schemes for nonlinear wave equations with variable coefficients

Abstract

In this article, we develop and analyze two energy-preserving high-order average vector field (AVF) compact finite difference schemes for solving variable coefficient nonlinear wave equations with periodic boundary conditions. Specifically, we first consider the variable coefficient nonlinear wave equation as an infinite-dimensional Hamiltonian system. Then the fourth-order compact finite difference and AVF techniques are applied to the resulting Hamiltonian system for the spatial discretization and time integration, respectively, which yield two energy-preserving high-order schemes, one is a time second-order AVF compact finite difference scheme (AVF(2)-CFD) and the other is a time fourth-order AVF compact finite difference scheme (AVF(4)-CFD). We theoretically prove that the proposed schemes satisfy energy conservations in the discrete forms and are uniquely solvable. Also, we prove the convergence of the proposed schemes and their error estimates, where the AVF(4)-CFD scheme is of fourth-order convergence in both time and space. Numerical experiments for the nonlinear wave equations with various nonlinearities are given to show their performances, which confirm the theoretical results.