# Radial basis function-generated finite differences with Bessel weights for the 2D Helmholtz equation

## TL;DR

This paper introduces a novel RBF-FD method using Bessel function weights to solve the 2D Helmholtz equation, effectively reducing dispersion and pollution effects in numerical solutions.

## Contribution

The paper presents a new RBF-FD scheme with Bessel weights and a regularization technique to improve stability and accuracy for the 2D Helmholtz equation.

## Key findings

- Mitigates dispersion effects in numerical solutions
- Controls condition number via regularization
- Demonstrates convergence and efficiency through numerical tests

## Abstract

In this paper we obtain approximated numerical solutions for the 2D Helmholtz equation using a radial basis function-generated finite difference scheme (RBF-FD), where weights are calculated by applying an oscillatory radial basis function given in terms of Bessel functions of the first kind. The problem of obtaining weights by local interpolation is ill-conditioned; we overcome this difficulty by means of regularization of the interpolation matrix by perturbing its diagonal. The condition number of this perturbed matrix is controlled according to a prescribed value of a regularization parameter. Different numerical tests are performed in order to study convergence and algorithmic complexity. As a result, we verify that dispersion and pollution effects are mitigated.