Fourier Beyond Dispersion: Wavenumber Explicit and Precise Accuracy of FDMs for the Helmholtz Equation

TL;DR

This paper introduces a Fourier analysis-based tool that accurately evaluates the convergence and error of finite difference methods for the Helmholtz equation, providing explicit wavenumber-dependent accuracy insights.

Contribution

It presents a novel, practical Fourier analysis tool that explicitly relates FDM accuracy to wavenumber, bridging the gap between dispersion analysis and actual error measurement.

Findings

Provides wavenumber explicit order of convergence

Enables rigorous proof of FDM accuracy for Helmholtz

Bridges dispersion analysis with real error assessment

Abstract

We propose a practical tool for evaluating and comparing the accuracy of FDMs for the Helmholtz equation. The tool based on Fourier analysis makes it easy to find wavenumber explicit order of convergence, and can be used for rigorous proof. It fills in the gap between the dispersion analysis and the actual error with source term.