Artificial boundary conditions for diffusion equations: Numerical study
type: Item
instance of: scholarly article
title: Artificial boundary conditions for diffusion equations: Numerical study
zbMATH Open document ID: 0873.65096
DOI: 10.1016/0377-0427(95)00140-9
author: Eric Dubach
published in: Journal of Computational and Applied Mathematics
publication date: 1997-10-30T00:00:00Z
review text: The paper is devoted to the numerical study of the advection-diffusion equation \[ \partial_t u+ (v\cdot \nabla) u-\nu \Delta u=0, \] where \(\nu\) is the viscosity (constant) and \(v= (v_1, v_2)\) is the advection velocity. In order to solve this equation in the disk \(D(R)= \{|X |< R\}\) with initial data having a compact support in \(D(R)\) the author considers the above equation in polar coordinates \(r,\theta\). Then \(\partial_ru\) on \(\partial D\) are determined by \(u\) on \(\partial D\) by the Dirichlet-to-Neumann operator \(N\), namely \(\partial_r u= -Nu\) (so-called transparent boundary condition). In order to compute \(N\) the author studies the symbol of \(N\) and approximates locally the operators represented in it. For doing this it is necessary to interpolate the function \(z^{1/2}\) by rational fractions in \(\mathbb{C} \setminus(-\infty, 0)\). Introducing suitably choosen interpolation points the author obtains explicitly the artificial boundary condition on \(\partial D\). Next, he takes \(v=0\), the Gaussian curve as initial data and \(u= u(t,r)\) for \(R\geq 2\). (Now interpolation points are choosen on the imaginary axis). He then tests the method numerically obtaining good results (compared with the exact solution) in the disk \(D(2)\). In the last section the author comes back to the case when \(v\) is not zero with \(u(X,0)= \exp (-\gamma |X |^2)\) \((\gamma>0)\) and again checks numerically the method.
Mathematics Subject Classification ID: 65M70
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 908434
zbMATH Keywords: advection-diffusion equation
zbMATH Keywords: artificial boundary conditions
zbMATH Keywords: collocation
reviewed by: S. Burys
MaRDI profile type: Publication
full work available at URL: 0377-0427(95)00140-9
OpenAlex ID: W2071618161
cites work: Absorbing Boundary Conditions for the Numerical Simulation of Waves
cites work: Absorbing boundary conditions for diffusion equations
cites work: Well-Posedness of One-Way Wave Equations and Absorbing Boundary Conditions
cites work: Q5549989
description: scientific article; zbMATH DE number 908434