A time-domain preconditioner for the Helmholtz equation

TL;DR

This paper introduces a novel time-domain preconditioner for the Helmholtz equation that improves solution accuracy and efficiency, especially for discretizations where traditional time-domain methods face limitations.

Contribution

The work develops a new preconditioning approach based on a matrix recurrence relation, enabling effective time-domain solutions for the Helmholtz equation with compact-stencil discretizations.

Findings

Achieves accurate approximate solutions through iterative recurrence.

Enhances GMRES convergence with a smooth window function.

Requires less memory and offers reasonable computation times.

Abstract

Time-harmonic solutions to the wave equation can be computed in the frequency or in the time domain. In the frequency domain, one solves a discretized Helmholtz equation, while in the time domain, the periodic solutions to a discretized wave equation are sought, e.g. by simulating for a long time with a time-harmonic forcing term. Disadvantages of the time-domain method are that the solutions are affected by temporal discretization errors and that the spatial discretization cannot be freely chosen, since it is inherited from the time-domain scheme. In this work we address these issues. Given an indefinite linear system satisfying certain properties, a matrix recurrence relation is constructed, such that in the limit the exact discrete solution is obtained. By iterating a large, finite number of times, an approximate solution is obtained, similarly as in a time-domain method for the Helmholtz equation. To improve the convergence, the process is used as a preconditioner for GMRES, and the time-harmonic forcing term is multiplied by a smooth window function. The construction is applied to a compact-stencil finite-difference discretization of the Helmholtz equation, for which previously no time-domain solver was available. Advantages of the resulting solver are the relative simplicity, small memory requirement and reasonable computation times.