2026/05/20 by Ansgar Jüngel, Panchi Li, Zhiwei Sun +1
#math.NA #cs.NA
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 \(kh∉π\mathbb Z\). Under the fixed-resolution condition \(kh≤ s0<π\) and \(kL≥π\), 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.