Fourier Analysis of Finite Difference Schemes for the Helmholtz Equation in 1D with Dirichlet  Conditions: Sharp Estimates and Relative Errors

Fourier Analysis of Finite Difference Schemes for the Helmholtz Equation in 1D with Dirichlet Conditions: Sharp Estimates and Relative Errors | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Fourier Analysis of Finite Difference Schemes for the Helmholtz Equation in 1D with Dirichlet Conditions: Sharp Estimates and Relative Errors Martin Gander, Hui Zhang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-7917103/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract We propose an approach based on Fourier analysis to wavenumber explicit sharp estimation of absolute and relative errors of finite difference methods for the Helmholtz equation {in 1D with Dirichlet boundary conditions} and general source terms. We use the approach to analyze the classical centred scheme. For the Fourier interpolants of the discrete solution with homogeneous (or inhomogeneous) Dirichlet conditions, we show rigorously that the worst case attainable convergence order of the absolute error is $k^2h^2$ (or $k^3h^2$) in the $L^2$-norm and $k^3h^2$ (or $k^4h^2$) in the $H^1$-semi-norm, and that of the relative error is $k^3h^2$ in both $L^2$- and $H^1$-semi-norms. Even though the classical centred scheme is well-known, it is the first time that such sharp estimates of absolute and relative errors are obtained. We show also that the Fourier analysis approach can be used as a convenient visual tool for evaluating finite difference schemes in presence of source terms, which is beyond the scope of dispersion analysis. finite difference Helmholtz equation error estimate wavenumber Fourier analysis Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 24 Feb, 2026 Reviews received at journal 24 Feb, 2026 Reviewers agreed at journal 24 Feb, 2026 Reviewers agreed at journal 18 Nov, 2025 Reviewers invited by journal 07 Nov, 2025 Editor assigned by journal 07 Nov, 2025 Submission checks completed at journal 07 Nov, 2025 First submitted to journal 21 Oct, 2025 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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