Radial basis function collocation method for decoupled fractional Laplacian wave equations

TL;DR

This paper introduces a radial basis function collocation method for solving decoupled fractional Laplacian wave equations, effectively handling irregular domains in seismic wave modeling, with demonstrated convergence and stability.

Contribution

It presents a novel RBF collocation approach that operates in the physical domain, overcoming limitations of Fourier pseudospectral methods for irregular geometries.

Findings

Method achieves stable long-time simulations.

Effective in both regular and irregular geometries.

Numerical results confirm convergence and stability.

Abstract

Decoupled fractional Laplacian wave equation can describe the seismic wave propagation in attenuating media. Fourier pseudospectral implementations, which solve the equation in spatial frequency domain, are the only existing methods for solving the equation. For the earth media with curved boundaries, the pseudospectral methods could be less attractive to handle the irregular computational domains. In the paper, we propose a radial basis function collocation method that can easily tackle the irregular domain problems. Unlike the pseudospectral methods, the proposed method solves the equation in physical variable domain. The directional fractional Laplacian is chosen from varied definitions of fractional Laplacian. Particularly, the vector Gr\"unwald-Letnikov formula is employed to approximate fractional directional derivative of radial basis function. The convergence and stability of the method are numerically investigated by using the synthetic solution and the long-time simulations, respectively. The method's flexibility is studied by considering homogeneous and multi-layer media having regular and irregular geometric boundaries.