An alternating-direction iteration method for Helmholtz problems
type: Item
instance of: scholarly article
title: An alternating-direction iteration method for Helmholtz problems
zbMATH Open document ID: 0807.65106
author: Jim jun. Douglas
author: Jeffrey L. jun. Hensley
author: Jean E. Roberts
published in: Applications of Mathematics
publication date: 1994-04-10T00:00:00Z
full work available at URL: https://eudml.org/doc/15756
review text: 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.
reviewed by: Kurt Frischmuth
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 519752
zbMATH Keywords: noncoercive nonsymmetric problems
zbMATH Keywords: Helmholtz equation
zbMATH Keywords: finite difference
zbMATH Keywords: alternating-direction iteration method
zbMATH Keywords: time-stepping method
zbMATH Keywords: convergence
zbMATH Keywords: numerical examples
MaRDI profile type: Publication
cites work: Alternating direction methods for three space variables
cites work: Q5649968
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cites work: The Numerical Solution of Parabolic and Elliptic Differential Equations
cites work: On convergence of alternating direction procedures
OpenAlex ID: W2600389761
DOI: 10.21136/AM.1993.104557
description: scientific article; zbMATH DE number 519752