Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Hui Zhang

TL;DR

The paper tackles the pollution effect in high-frequency Helmholtz discretizations by introducing a Fourier-analysis framework that yields wavenumber-explicit error bounds for finite-difference methods. It develops a frequency-domain decomposition based on discrete and continuous symbols, defines a sharp error measure $ψ_p$ to bound $L^2$ and $H^1$ errors, and analyzes classical as well as dispersion-free high-order schemes in 1D, with extensions to higher dimensions planned. Theoretical results show that, for an $m$th-order scheme, the $L^2$-error scales with $h^m$ (and is influenced by the wavenumber through the source regularity) while the $H^1$-error scales with $k^m h^m$ for dispersion-free schemes, and with $k$-dependent amplification near the physical wavenumber $k$ for traditional schemes. Numerical experiments corroborate the predicted $k$- and $h$-scaling, demonstrating the advantage of dispersion-free schemes in mitigating pollution and providing a practical, rigorous tool for evaluating and designing FDMs for Helmholtz problems, with clear path to 2D/3D extensions.

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.

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

TL;DR

The paper tackles the pollution effect in high-frequency Helmholtz discretizations by introducing a Fourier-analysis framework that yields wavenumber-explicit error bounds for finite-difference methods. It develops a frequency-domain decomposition based on discrete and continuous symbols, defines a sharp error measure to bound and errors, and analyzes classical as well as dispersion-free high-order schemes in 1D, with extensions to higher dimensions planned. Theoretical results show that, for an th-order scheme, the -error scales with (and is influenced by the wavenumber through the source regularity) while the -error scales with for dispersion-free schemes, and with -dependent amplification near the physical wavenumber for traditional schemes. Numerical experiments corroborate the predicted - and -scaling, demonstrating the advantage of dispersion-free schemes in mitigating pollution and providing a practical, rigorous tool for evaluating and designing FDMs for Helmholtz problems, with clear path to 2D/3D extensions.

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.

Paper Structure

This paper contains 5 sections, 1 theorem, 15 equations, 3 figures.

Table of Contents

  1. Introduction.
  2. Analysis in 1D
  3. FDMs in 1D and their symbols
  4. Accuracy of FDMs in 1D
  5. Numerical experiments in 1D

Key Result

Theorem 1

Suppose $k\not\in\pi\mathbb{Z}$, $f\in H^p_0(0,1)$ and $f$ is band limited with the Fourier coefficients $\hat{f}_n=0$ for $n\ge N$ for an even integer $N\ge 4$. Let $u$ be the solution of helm1d. Suppose a linear finite difference equation $\mathcal{H}^hu^h=\mathcal{R}^hf$ on $\Omega_h$ with $u^h(0 where $|v|_{H^p(0,1)}=\|v^{(p)}\|_{L^2(0,1)}$ is the $H^p$-semi-norm, $\xi_n=n\pi$, and both upper

Figures (3)

  • Figure 1: Symbol errors in 1D: $k^p\psi_p$ for $L^2$-norm (col.1) and $k^p\psi_{p-1}$ for $H^1$-semi-norm (col.2).
  • Figure 2: Symbol errors in 1D with $kh$ fixed and $k$ increases.
  • Figure 3: Numerical errors in 1D: monochromatic $k$-dependent source (col.1-2), mixed and fixed source (col.3-4).

Theorems & Definitions (3)

  • Theorem 1
  • proof
  • Remark 1