Fourier pseudospectral methods for the spatial variable-order fractional wave equations
CoRR(2023)
摘要
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 𝒪(Nlog
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 𝒪(MNlog N) and storage cost 𝒪(MN), where M ≪ 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.
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