In this paper, we propose Fourier pseudospectral methods to solve the variable-order space fractional wave equation and develop an accelerated matrix-free approach for its effective implementation. In constant-order cases, our methods can be efficiently implemented via the (inverse) fast Fourier transforms, and the computational cost at each time step is ${\mathcal O}(N\log N)$ with $N$ the total number of spatial points. However, this fast algorithm fails in the variable-order cases due to the spatial dependence of the Fourier multiplier. On the other hand, the direct matrix-vector multiplication approach becomes impractical due to excessive memory requirements. To address this challenge, we proposed an accelerated matrix-free approach for the efficient computation of variable-order cases. The computational cost is ${\mathcal O}(MN\log N)$ and storage cost ${\mathcal O}(MN)$, where $M \ll N$. Moreover, our method can be easily parallelized to further enhance its efficiency. Numerical studies show that our methods are effective in solving the variable-order space fractional wave equations, especially in high-dimensional cases. Wave propagation in heterogeneous media is studied in comparison to homogeneous counterparts. We find that wave dynamics in fractional cases become more intricate due to nonlocal interactions. Specifically, dynamics in heterogeneous media are more complex than those in homogeneous media.

翻译：本文提出傅里叶伪谱方法求解变阶空间分数阶波动方程，并发展了一种加速无矩阵方法以实现高效计算。在常阶情形下，该方法可通过（逆）快速傅里叶变换高效实现，每时间步计算复杂度为${\mathcal O}(N\log N)$，其中$N$为空间网格点数。然而，由于变阶情形中傅里叶乘子具有空间依赖性，该快速算法失效。另一方面，直接矩阵-向量乘法因内存需求过大而难以实际应用。为解决这一挑战，我们提出了一种加速无矩阵方法用于变阶情形的高效计算，其计算复杂度为${\mathcal O}(M N\log N)$，存储复杂度为${\mathcal O}(MN)$，其中$M \ll N$。此外，该方法易于并行化以进一步提升效率。数值研究表明，该方法能有效求解变阶空间分数阶波动方程，尤其在高维情形中表现优异。本文还对比研究了波在非均匀介质与均匀介质中的传播特性，发现分数阶情形下的波动力学因非局部相互作用而更为复杂——具体而言，非均匀介质中的动力现象较均匀介质更为复杂。