A Robust Adaptive Method for a Quasi-Linear One-Dimensional Convection-Diffusion Problem

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 ε, where ε is the diffusion coefficient. In the case when the boundary value problem is linear, if N is sufficiently large independently of ε, it is shown that after O( ln(1/ε)/ ln N) iterations of the algorithm, the piecewise linear interpolant of the computed solution achieves first-order accuracy in the L^∞[0,1] norm uniformly in ε. Numerical experiments are presented that support our theoretical results.