A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

TL;DR

This paper introduces a multigrid solver for the Helmholtz equation with a point source, leveraging a reformulation into travel time and amplitude equations to improve efficiency and accuracy at high wave numbers.

Contribution

The paper presents a novel multigrid method based on reformulating the Helmholtz equation into travel time and amplitude equations, enhancing solution efficiency and accuracy.

Findings

Second order ADR discretization outperforms standard methods at high wave numbers.

The approach is more efficient than the shifted Laplacian multigrid method.

Amplitude solutions are smoother and better suited for multigrid techniques.

Abstract

The Helmholtz equation arises when modeling wave propagation in the frequency domain. The equation is discretized as an indefinite linear system, which is difficult to solve at high wave numbers. In many applications, the solution of the Helmholtz equation is required for a point source. In this case, it is possible to reformulate the equation as two separate equations: one for the travel time of the wave and one for its amplitude. The travel time is obtained by a solution of the factored eikonal equation, and the amplitude is obtained by solving a complex-valued advection-diffusion-reaction (ADR) equation. The reformulated equation is equivalent to the original Helmholtz equation, and the differences between the numerical solutions of these equations arise only from discretization errors. We develop an efficient multigrid solver for obtaining the amplitude given the travel time, which can be efficiently computed. This approach is advantageous because the amplitude is typically smooth in this case, and hence, more suitable for multigrid solvers than the standard Helmholtz discretization. We demonstrate that our second order ADR discretization is more accurate than the standard second order discretization at high wave numbers, as long as there are no reflections or caustics. Moreover, we show that using our approach, the problem can be solved more efficiently than using the common shifted Laplacian multigrid approach.