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.

在本文中，我们开发和分析了两种节能的高阶平均矢量场（AVF）紧致有限差分方案，用于求解具有周期性边界条件的变系数非线性波动方程。具体而言，我们首先将变系数非线性波动方程视为一个无限维哈密顿系统。然后将四阶紧致有限差分和AVF技术分别应用于所得的哈密顿系统进行空间离散化和时间积分，产生两个能量守恒的高阶方案，一种是时间二阶AVF紧致有限差分。方案（AVF（2）-CFD），另一种是时间四阶AVF紧凑有限差分方案（AVF（4）-CFD）。我们从理论上证明了所提出的方案满足离散形式的能量守恒并且是唯一可解的。此外，我们证明了所提方案及其误差估计的收敛性，其中AVF（4）-CFD方案在时间和空间上均为四阶收敛。给出了具有各种非线性的非线性波动方程的数值实验，以证明它们的性能，证实了理论结果。