Absorbing boundary conditions for diffusion equations
type: Item
instance of: scholarly article
title: Absorbing boundary conditions for diffusion equations
zbMATH Open document ID: 0838.65117
published in: Numerische Mathematik
publication date: 1996-06-03T00:00:00Z
full work available at URL: http://hdl.handle.net/2027.42/41953
review text: In dealing numerically with problems for unbounded regions, such as radiation problems, the reduction to bounded regions with artificial boundary conditions has received much attention, particularly in the hyperbolic case. The present paper deals with such reductions for diffusion equations \(Lu = 0\) (the heat equation \(\partial_t u - \Delta u = 0\) being a typical example) in a region with a smooth boundary \(\mathbb{R}_t \times S\). The Neumann operator \(N\), which yields the exterior normal derivative \(\partial u/ \partial v = - Nu\) on the boundary in terms of the boundary data of \(u\) itself for a solution of the equation, is studied. Its properties as a pseudodifferential operator --- in particular the leading terms of its symbol --- are discussed. For the heat operator \(N = (\partial_t - \Delta_S)^{1/2}\), \(\Delta_S\) being the Laplacian on \(S\). In general the actual computation of \(Nu\) is a problem from the numerical point of view. Construction of simple approximants are suggested, using only one or two leading terms of the symbol and rational approximants \[ z^{1/2} \approx \alpha z + \beta + \sum^m_{k = 1} \alpha_k z/(z + d_k), \] where, for the sake of stability, \(\alpha \geq 0\), \(\beta \geq 0\), \(\alpha_k > 0\), \(d_k > 0\). One of the two alternatives which are studied is \[ (\partial_t - \Delta_S)^{1/2} \approx \alpha (\partial_t - \Delta_S) + \sum^m_{k = 1} \alpha_k (\partial_t - \Delta_S) (\partial_t - \Delta_S + d_k)^{-1}. \] The existence of rational approximations of the type given above are discussed. No convergence proof is given but it is shown that the approximation has a smoothing property.
Mathematics Subject Classification ID: 65N45
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 41A20
zbMATH DE Number: 802461
zbMATH Keywords: absorbing boundary conditions
zbMATH Keywords: unbounded regions
zbMATH Keywords: radiation problems
zbMATH Keywords: artificial boundary conditions
zbMATH Keywords: diffusion equations
zbMATH Keywords: pseudodifferential operator
zbMATH Keywords: rational approximants
zbMATH Keywords: stability
author: Laurence Halpern
author: Jeffrey Rauch
reviewed by: S. Leif Svensson
MaRDI profile type: Publication
OpenAlex ID: W2141613295
DOI: 10.1007/S002110050141
description: scientific article; zbMATH DE number 802461