Meshless RBF based pseudospectral solution of acoustic wave equation

TL;DR

This paper introduces a meshless radial basis function pseudospectral method for solving the acoustic wave equation, overcoming polynomial basis limitations and enabling irregular grid use for multivariate problems.

Contribution

It presents a novel RBF-PS algorithm for inhomogeneous Helmholtz equations, demonstrating its advantages over traditional PS and finite difference methods.

Findings

RBF-PS achieves comparable accuracy to polynomial PS methods.

The method effectively handles irregular grids in multivariate systems.

Successful application to frequency domain acoustic wave propagation.

Abstract

Chebyshev pseudospectral (PS) methods are reported to provide highly accurate solution using polynomial approximation. Use of polynomial basis functions in PS algorithms limits the formulation to univariate systems constraining it to tensor product grids for multi-dimensions. Recent studies have shown that replacing the polynomial by radial basis functions in pseudospectral method (RBF-PS) has the advantage of using irregular grids for multivariate systems. A RBF-PS algorithm has been presented here for the numerical solution of inhomogeneous Helmholtz's equation using Gaussian RBF for derivative approximation. Efficacy of RBF approximated derivatives has been checked through error analysis comparison with PS method. Comparative study of PS, RBF-PS and finite difference approach for the solution of a linear boundary value problem has been performed. Finally, a typical frequency domain acoustic wave propagation problem has been solved using Dirichlet boundary condition and a point source. The algorithm presented here can be extended further for seismic modeling with complexities associated with absorbing boundary conditions.