Remarks on the wave-ray multigrid solvers for Helmholtz equations

Abstract

This chapter describes a “wave-ray” algorithm for solving the Helmholtz equations with radiation boundary conditions, and discusses the measures needed to be taken to make the solver efficient and accurate. An algebraic multigrid (AMG) solver for Helmholtz eigenvalue problems with variable potentials—is also introduced in the chapter. The multigrid approach is beneficial if the Helmholtz equations are considered on large (infinite) domain and accompanied by the radiation boundary conditions. The quality of the numerical solution in most multigrid solvers is controlled by the properties of the finest, target grid while coarse grids are used to provide a smooth correction to the fine-grid solution. In the case of Helmholtz equations, the most significant impact on the solution accuracy is a phase error—the error that occurs because of the accumulated differences between the wavelengths of differential and corresponded discrete principal components. Two other types of errors may decrease the solution accuracy—errors introduced by interpolation of the boundary conditions from coarse ray grids, and errors caused by the reflection from artificial boundaries that also occur on the coarsest ray grids.