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

摘要

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