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

TL;DR

This work develops a Fourier pseudospectral framework for variable-order space fractional wave equations, addressing the computational bottleneck posed by the spatially varying operator $(-Δ)^{s(x)}$ via an accelerated matrix-free approach. By expanding the Fourier multiplier around an average order $s_0$ and truncating to $M$ terms, the method achieves ${\mathcal O}(M N\log N)$ time and ${\mathcal O}(MN)$ storage, avoiding dense matrices while retaining accuracy. Three second-order temporal schemes—CNFP (implicit), LFFP (explicit), and TSFP2 (explicit, Strang splitting)—are analyzed and demonstrated to be accurate and efficient, with TSFP2 offering a favorable balance of ease of implementation and stability. Numerical experiments reveal that heterogeneity in $s(x)$ leads to more complex, asymmetric wave dynamics compared with homogeneous media, and the matrix-free approach substantially lowers memory and computation costs, especially in higher dimensions. The methods are versatile and scalable, with potential applications to seismic wave propagation and other variable-order fractional problems.

Abstract

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.

Figures (10)

• Figure 1: (a) Illustration of the relation between $|e(M, \mu_k)|$, $M$ and $\mu_k$, where $s_0 = 1$ is used. (b) The $l_2$-norm errors in approximating function $(-\Delta)^{s(x)}u(x)$ for different $h$ and $M$, where $s(x) = 1+0.3\sin(\pi x/8)$ and $u(x)$ is defined in (\ref{['example3']}).
• Figure 2: Critical time step $\tau^*$ versus mesh size $h$ for LFFP method, where $s_1(x) = 1+0.3\sin(\pi x/8)$, and $s_2(x) = 1+0.2\tanh(\cos(\pi x/8))$.
• Figure 3: Critical time step $\tau^*$ versus mesh size $h$ for TSFP2 method, where $s_1(x) = 1+0.3\sin(\pi x/8)$, and $s_2(x) = 1+0.2\tanh(\cos(\pi x/8))$.
• Figure 4: Solution dynamics of the nonlinear wave equation in (\ref{['example1']}) with different $s(x)$. For better illustration, the displayed domain $[-15, 15]$ is much smaller than our computational domain.
• Figure 5: Comparison of the matrix-free approach (symbol '$+$') and the direct matrix-vector approach (symbol '$\circ$') in solving the problem (\ref{['example1']}) with $s(x) = 1+0.3\sin(\pi x/8)$, where the TSFP2 method is used with $\tau = 0.0001$. (a) Numerical errors at time $t = 1$; (b) Computing time taken from $t = 0$ to $t = 1$.

Theorems & Definitions (1)

• Remark 3.1