Optimal difference schemes on piecewise‐uniform meshes for a singularly perturbed parabolic convection‐diffusion equation

Abstract

A grid approximation of a boundary value problem is considered for a singularly perturbed parabolic convection‐diffusion equation. For this problem, upwind difference schemes on the well‐known piecewise‐uniform meshes converge ϵ‐uniformly in the maximum discrete norm at the rate O(N^{− 1}lnN + N₀ ⁻¹ ), where N + 1 and N ₀ + 1 are the number of mesh points in x and t respectively; the number of nodes in the x‐mesh before the transition point (the point where the step‐size changes) and after it are the same. Under the condition N Â N ₀ this scheme converges at the rate O(P^−1/2 ln P); here P = (N + 1)(N ₀ + 1) is the total number of nodes in the piecewise‐uniform mesh. Schemes on piecewise‐uniform meshes are constructed that are optimal with respect to the convergence rate. These schemes converge ϵ‐uniformly at the rate O(P^−1/2 ln^1/2 P). In optimal meshes based on widths that are similar to Kolmogorov's widths, the ratio of mesh points in x and t is of O((ϵ + ln⁻¹ P)⁻¹). Under the condition ϵ = o( 1), most nodes in such a mesh in x are placed before the transition point.

Key words:

• boundary value problem
• perturbation parameter ϵ
• parabolic convection‐diffusion equation
• finite difference approximation
• optimal meshes
• boundary layer
• Kolmogorov's widths
• ϵ‐uniform convergence