Paper Info
Title
A Robust Adaptive Method for a Quasi-Linear One-Dimensional Convection-Diffusion Problem
Abstract
A quasi-linear conservative convection-diffusion two-point boundary value problem is considered. To solve it numerically, an upwind finite difference scheme is applied. The mesh used has a fixed number (N+1) of nodes and is initially uniform, but its nodes are moved adaptively using a simple algorithm of de Boor based on equidistribution of the arc-length of the current computed piecewise linear solution. It is proved for the first time that a mesh exists that equidistributes the arc-length along the polygonal solution curve and that the corresponding computed solution is first-order accurate, uniformly in $\varepsilon$, where $\varepsilon$ is the diffusion coefficient. In the case when the boundary value problem is linear, if N is sufficiently large independently of $\varepsilon$, it is shown that after $O({\rm ln}(1/\varepsilon)/{\rm ln} N)$ iterations of the algorithm, the piecewise linear interpolant of the computed solution achieves first-order accuracy in the $L^\infty[0,1]$ norm uniformly in $\varepsilon$. Numerical experiments are presented that support our theoretical results.
Year
DOI
Venue
2001
10.1137/S003614290138471X
SIAM Journal on Numerical Analysis
Keywords
Field
DocType
quasi-linear,conservative,convection-diffusion problem,upwind scheme,singular perturbation,adaptive mesh,equidistribution
Convection–diffusion equation,Boundary value problem,Mathematical analysis,De Boor's algorithm,Singular perturbation,Finite difference method,Upwind scheme,SIMPLE algorithm,Piecewise linear function,Mathematics
Journal
Volume
Issue
ISSN
39
4
0036-1429
Citations 
PageRank 
References 
37
3.79
3
Authors
2
Name
Order
Citations
PageRank
Natalia Kopteva113022.08
Martin Stynes227357.87