In this paper, we develop and analyze a new time fourth-order energy-preserving average vector field (AVF) finite difference method for the nonlinear fractional wave equations with Riesz space-fractional derivative. To the corresponding Hamiltonian system of the nonlinear fractional wave equations, the fourth-order weighted and shifted Lubich difference operator in space and the fourth-order AVF method in time are used to develop the time fourth-order energy-preserving AVF finite difference method for the nonlinear fractional wave equations. The energy conservation in the discrete form and unique solvability of the proposed scheme are proved and error estimates of the scheme are further proved to be order of $O({\left(\Delta t\right)}^4+h^4)$ in the discrete $L^2$- norm. Numerical experiments confirm energy conservation and high-order accuracy of the proposed scheme.

本文针对具有Riesz空间分数阶导数的非线性分数波方程，开发并分析了一种新的时间四阶能量守恒平均矢量场（AVF）有限差分方法。对于相应的非线性分数波方程的汉密尔顿系统，使用空间中的四阶加权移位Lubich差分算子和及时的四阶AVF方法来开发时间四阶能量守恒AVF有限差分方法。非线性分数波方程。证明了该方案的离散形式的能量节约和独特的可解性，并进一步证明了该方案的误差估计为阶次。$Ø（{（\Delta Ť）}^4+H^4）$ 在离散 ${\mathrm{大号}}^2$-规范。数值实验证实了该方案的节能性和高阶精度。