In this paper, we numerically investigate the space fractional nonlinear damped wave equation. We construct a novel high-accuracy dissipation-preserving finite difference scheme by using the new fourth-order fractional central difference operator. Thanks to the toeplitz-like differentiation matrix, we further raise the computation efficiency of the proposed scheme by fast Fourier transform. Moreover, we obtain the error estimate of our proposed scheme in L² and H^α/2 (1 < α ≤ 2) norm, respectively. Finally, we verify the discrete dissipation-preserving law and convergence of the proposed scheme by one- and two-dimensional numerical experiments in long-time observation.

在本文中，我们对空间分数非线性阻尼波方程进行了数值研究。通过使用新的四阶分数中心差分算子，构造了一种新颖的高精度耗散保持有限差分方案。得益于类似于toeplitz的微分矩阵，我们通过快速傅立叶变换进一步提高了所提出方案的计算效率。此外，我们得到我们所提出的方案的误差估计在大号²和ħ ^{α / 2}（1 <  α 分别≤2）范数。最后，通过长期观测的一维和二维数值实验，验证了所提方案的离散性保持规律和收敛性。