At the boundary of infinity: numerical methods for unbounded domains
type: Item
instance of: scholarly article
title: At the boundary of infinity: numerical methods for unbounded domains
zbMATH Open document ID: 0948.65103
author: Marcus J. Grote
published in: Elemente der Mathematik
publication date: 2000-11-26T00:00:00Z
review text: The paper is the revised text of the author's inaugural lecture, given in April 1998 at the ETH Zürich. It is devoted to the numerical analysis of physical problems in unbounded domains. The key idea for a solution of such a problem, e.g. the wave propagation in one or more dimensions, is the introduction of an artificial boundary and appropriate conditions on it. For the convergence of an approximate solution to the exact one, the discretization error as well as the error due to the artificial boundary have to be studied. In addition, an approximation has to be efficient, as the author shows by a simple example. The necessary steps for the solution of an unbounded domain problem are discussed for the one-dimensional linear wave equation. Afterwards, the author turns to the much more complicated multidimensional case, and pseudo-differential operators come into play in view of nonlocal boundary conditions. Alternatively, the author presents the idea of Kirchhoff's integral representation that is based upon the famous Huygens's principle. Finally, numerical tests illustrate different approaches and the inherent question of convergence. The paper gives quite a nice insight into this highly interesting but complicated field of research and is addressed to the non-specialist.
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 65M12
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 65M15
zbMATH DE Number: 1475282
zbMATH Keywords: error bounds
zbMATH Keywords: unbounded domains
zbMATH Keywords: wave propagation
zbMATH Keywords: artificial boundary
zbMATH Keywords: convergence
zbMATH Keywords: wave equation
zbMATH Keywords: pseudo-differential operators
zbMATH Keywords: nonlocal boundary conditions
zbMATH Keywords: Kirchhoff's integral representation
zbMATH Keywords: Huygens's principle
zbMATH Keywords: numerical tests
reviewed by: Etienne Emmrich
MaRDI profile type: Publication
full work available at URL: s000170050073
OpenAlex ID: W2041603895
cites work: Radiation boundary conditions for wave-like equations
cites work: A perfectly matched layer for the absorption of electromagnetic waves
cites work: Absorbing Boundary Conditions for the Numerical Simulation of Waves
cites work: Non-reflecting boundary conditions
DOI: 10.1007/S000170050073
description: scientific article; zbMATH DE number 1475282