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

Fourier pseudospectral methods for the spatial variable-order fractional wave equations