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.