A Bernoulli Phase-Fitted finite difference method with wavenumber-explicit analysis for the Helmholtz problem

Abstract

A new Bernoulli phase-fitted finite difference method for the Helmholtz equation is introduced, obtained by applying a complexified Scharfetter--Gummel flux to the one-way factors of the operator. The rigorous analysis is developed for the one-dimensional Helmholtz problem with impedance boundary conditions. For the homogeneous problem, the scheme reproduces sampled plane-waves exactly, both in the interior and at the discrete impedance boundary closures. For the inhomogeneous problem, we prove wavenumber-explicit stability, consistency, and second-order convergence estimates for all nondegenerate mesh wavenumbers \(kh\notinπ\mathbb Z\). Under the fixed-resolution condition \(kh\le s_0<π\) and \(kL\geπ\), the estimates yield a pollution-free convergence theory. Numerical experiments confirm the plane-wave exactness and the predicted convergence behavior, and show favorable fixed-resolution performance compared with standard and dispersion-corrected finite difference methods.