The RBF-FD method for solving the time-fractional viscoelastic wave propagation in irregular domains

Abstract

The time-fractional viscoelastic wave equation plays a crucial role in geophysical exploration by accurately modeling wave attenuation and velocity dispersion in Earth’s media. However, solving this equation is challenging due to the stress–strain relationship governed by the Caputo fractional derivative of small orders and the complexity of irregular surface topographies. The requirement for significant memory and computational resources when dealing with small fractional orders limits the efficiency of traditional methods. Conventional approaches, which rely on horizontal reference planes, fixed-step grids, and stair-step approximations for irregular surfaces, often lead to staircase scattering and reduced accuracy. To address these challenges, this study proposes a numerical algorithm based on the Radial Basis Function-Finite Difference (RBF-FD) method for simulating time-fractional viscoelastic waves in irregular domains. The meshless nature of the RBF-FD method allows for flexible node distribution, making it well-suited for complex interfaces. Additionally, a short-memory algorithm is implemented to efficiently solve the stress–strain relationship governed by the fractional derivative. Several numerical experiments are presented to validate the accuracy and efficiency of the proposed scheme.