Lightning Helmholtz Solver

TL;DR

This paper applies the Lightning Method to solve the Helmholtz equation in 2D exterior domains, providing insights on implementation, parameter selection, and solution behavior consistent with wave physics.

Contribution

It demonstrates the application of the Lightning Method to Helmholtz problems in exterior domains and offers practical guidance on implementation and parameter tuning.

Findings

Solutions decay to near zero in shadow regions

Method aligns with wave propagation theory

Guidelines for parameter selection in complex regions

Abstract

In this dissertation we have applied Trefethen and Gopal's Lightning Method to solve the Helmholtz equation in the exterior of two dimensional piecewise smooth domains. The background theory motivating the method is presented, and we explore the optimal method implementation for the unit square, which is subsequently used to give a guide on parameter selection for a general region. The behaviour of the computed solutions is verified to act in accordance with our intuition and current understanding of wave propagation, and we show that the wave decays to approximately 0 in the shadow region.