An alternating-direction iteration method for Helmholtz problems
A complex-valued Helmholtz problem is considered on a unit square. It describes waves under absorbing boundary conditions. A finite difference approximation on a uniform grid with centered first and second differences is used for the discretization. The aim of the paper is the efficient solution of the arising algebraic equations by an alternating- direction iteration method. In Section 2 the iteration is introduced as a time-stepping method for a corresponding evolution problem with a set of complex iteration parameters (reciprocals of pseudo time-steps). An analysis of the error shows that there is no single pseudo time-step giving decay for all modes, hence a cycle of different steps has to be used. In dependence of the eigenvalues of a tridiagonal matrix a set of suitable parameters can be given. In Section 3 it is shown that the above choice of parameters -- although ensuring convergence -- is unsatisfactory for practical calculations because of rounding problems. A heuristic modification of the method is tested and proved to yield ``rapid and effective convergence''. In Section 4 modifications for the case of attenuated waves are discussed. All results remain valid with the generalized friction terms included. The paper is concluded by numerical examples. The cases with and without attenuation are compared with each other and with one containing two source terms. Finally, traces of displacements at four receiver positions are given, and again the effect of attenuation is studied.
scientific article; zbMATH DE number 519752