Extensions and Analysis of an Iterative Solution of the Helmholtz Equation via the Wave Equation

TL;DR

This paper extends the analysis of the WaveHoltz iterative method for solving the Helmholtz equation, demonstrating convergence under various boundary conditions and discretizations, and showing how to eliminate time discretization errors.

Contribution

It provides new convergence proofs for the WaveHoltz iteration with impedance and damping conditions, and analyzes its discrete form with higher order time-stepping schemes.

Findings

Convergence of WaveHoltz for impedance boundary conditions in 1D.

Frequency-independent convergence with sufficient damping in any dimension.

Complete removal of time discretization error through discrete analysis.

Abstract

In this paper we extend analysis of the WaveHoltz iteration -- a time-domain iterative method for the solution of the Helmholtz equation. We expand the previous analysis of energy conserving problems and prove convergence of the WaveHoltz iteration for problems with impedance boundary conditions in a single spatial dimension. We then consider interior Dirichlet/Neumann problems with damping in any spatial dimension, and show that for a sufficient level of damping the WaveHoltz iteration converges in a number of iteration independent of the frequency. Finally, we present a discrete analysis of the WaveHoltz iteration for a family of higher order time-stepping schemes. We show that the fixed-point of the discrete WaveHoltz iteration converges to the discrete Helmholtz solution with the order of the time-stepper chosen. We present numerical examples and demonstrate that it is possible to completely remove time discretization error from the WaveHoltz solution through careful analysis of the discrete iteration together with updated quadrature formulas.