In this paper, we propose Fourier pseudospectral methods to solve variable-order fractional viscoacoustic wave equations. Our approach involves a Fourier pseudospectral method for spatial discretization and an accelerated matrix-free technique for efficient computation and storage costs, with a computational cost of $\mathcal{O}(MN\log N)$ and storage cost $\mathcal{O}(MN)$ where $M\ll N$. For temporal discretization, we employ the Crank-Nicolson, leap-frog, and time-splitting schemes. Numerical experiments are conducted to assess their performance. The results demonstrate the advantages of our fast method, particularly in computational and storage costs, and its feasibility in high dimensions. The numerical findings reveal that all three temporal discretization methods exhibit second-order accuracy, while the Fourier pseudospectral spatial discretization showcases spectral accuracy.

翻译：本文提出傅里叶伪谱方法求解变阶分数阶黏声波方程。我们的方法包含空间离散的傅里叶伪谱技术与加速无矩阵计算技术，可有效降低计算与存储成本，其中计算复杂度为$\mathcal{O}(MN\log N)$，存储复杂度为$\mathcal{O}(MN)$（$M\ll N$）。时间离散采用Crank-Nicolson格式、蛙跳格式和分裂时间格式。数值实验评估了各方法的性能，结果表明所提出的快速方法在计算与存储成本方面具有显著优势，且在高维问题中具有可行性。数值结果揭示，三种时间离散方法均达到二阶精度，而傅里叶伪谱空间离散展现出谱精度。