A stabilized local integral method using RBFs for the Helmholtz equation with applications to wave chaos and dielectric microresonators

TL;DR

This paper introduces a stabilized local integral boundary domain method using Radial Basis Functions for solving the Helmholtz equation, demonstrating improved accuracy and stability in complex wave-related applications.

Contribution

It presents a novel stabilized RBF-based boundary integral method that enhances solution stability and accuracy for Helmholtz problems in irregular geometries.

Findings

Achieves high accuracy with small shape parameters in RBF interpolation.

Demonstrates improved stability over traditional RBF methods.

Effectively applies to wave chaos and dielectric microresonator problems.

Abstract

Most problems in electrodynamics do not have an analytical solution so much effort has been put in the development of numerical schemes, such as the finite-difference method, volume element methods, boundary element methods, and related methods based on boundary integral equations. In this paper we introduce a local integral boundary domain method with a stable calculation based on Radial Basis Functions (RBF) approximations, in the context of wave chaos in acoustics and dielectric microresonator problems. RBFs have been gaining popularity recently for solving partial differential equations numerically, becoming an extremely effective tool for interpolation on scattered node sets in several dimensions with high-order accuracy and flexibility for nontrivial geometries. One key issue with infinitely smooth RBFs is the choice of a suitable value for the shape parameter which controls the flatness of the function. It is observed that best accuracy is often achieved when the shape parameter tends to zero. However, the system of discrete equations obtained from the interpolation matrices becomes ill-conditioned, which imposes severe limits to the attainable accuracy. A few numerical algorithms have been presented that are able to stably compute an interpolant, even in the increasingly flat basis function limit. We present the recently developed Stabilized Local Boundary Domain Integral Method in the context of boundary integral methods that improves the solution of the Helmholtz equation with RBFs. Numerical results for small shape parameters that stabilize the error are shown. Accuracy and comparison with other methods are also discussed for various case studies. Applications in wave chaos, acoustics and dielectric microresonators are discussed to showcase the virtues of the method, which is computationally efficient and well suited to the kind of geometries with arbitrary shape domains.