A Bernoulli phase-fitted finite difference method and wavenumber-explicit analysis for the one-dimensional Helmholtz equation

TL;DR

This paper introduces a Bernoulli phase-fitted finite difference method for the Helmholtz equation that is exact for plane waves and provides wavenumber-explicit stability and pollution-free convergence.

Contribution

The authors develop a novel phase-fitted finite difference scheme based on complexified Scharfetter--Gummel discretization, ensuring minimal dispersion and artificial reflection, with rigorous stability and convergence analysis.

Findings

Method is exact for plane waves, eliminating numerical dispersion.

Provides wavenumber-explicit stability estimates.

Achieves pollution-free convergence under certain conditions.

Abstract

We propose a Bernoulli phase-fitted (BPF) finite difference method for the Helmholtz equation on the interval $(0, L)$ with impedance boundary conditions. The scheme is derived from a complexified Scharfetter--Gummel discretization of the one-way factorization of the Helmholtz operator. It yields both a phase-fitted interior discretization and exact discrete impedance boundary closures. For the homogeneous problem, the method is exact for plane waves, so the scheme introduces neither numerical dispersion in the interior nor artificial reflection at the boundaries. For the inhomogeneous problem, we prove well-posedness, derive wavenumber-explicit stability estimates, and establish second-order consistency and convergence valid for all $kh\notin\pi\mathbb Z$, where $k$ is the wavenumber and $h$ the grid size. In particular, under the fixed-resolution condition $kh\le s_0$ for some $0<s_0<\pi$ together with $kL\ge\pi$, the constants in the error bounds remain uniform with respect to the wavenumber, yielding a pollution-free convergence theory in the principal Nyquist regime. Numerical experiments confirm the theoretical analysis and show favorable performance compared with standard and dispersion-corrected finite difference methods.