In convection-diffusion problems, transport processes dominate while diffusion effects are confined to a relatively small part of the domain. This state of affairs means that one cannot rely on the formal ellipticity of the differential operator to ensure the convergence of standard numerical algorithms. Thus new ideas and approaches are required.The survey begins by examining the asymptotic nature of solutions to stationary convection-diffusion problems. This provides a suitable framework for the understanding of these solutions and the difficulties that numerical techniques will face. Various numerical methods expressly designed for convection-diffusion problems are then presented and extensively discussed. These include finite difference and finite element methods and the use of special meshes.

Steady-state convection-diffusion problems

A brief survey on numerical methods for solving singularly perturbed problems

A coupled system of two singularly perturbed linear reaction-diffusion two-point boundary value problems is examined. The leading term of each equation is multiplied by a small positive parameter, but these parameters may have different magnitudes. The solutions to the system have boundary layers that overlap and interact. The structure of these layers is analysed, and this leads to the construction of a piecewise-uniform mesh that is a variant of the usual Shishkin mesh. On this mesh central differencing is proved to be almost first-order accurate, uniformly in both small parameters. Supporting numerical results are presented for a test problem.

A uniformly convergent numerical method for a coupled system of two singularly perturbed linear reaction–diffusion problems

Layer-adapted meshes for convection-diffusion problems

A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations

Introduction to Numerical Methods for Problems with Boundary Layers Numerical Methods on Uniform Meshes Layer Resolving Methods for Convection-Diffusion Problems in One Dimension The Limitations of Non-Monotone Numerical Methods Convection-Diffusion Problems in a Moving Medium Convection-Diffusion Problems with Frictionless Walls Convection-Diffusion Problems with No Slip Boundary Conditions Experimental Estimation of Errors Non-Monotone Methods in Two Dimensions Linear and Nonlinear Reaction-Diffusion Problems Prandtl Flow past a Flat Plate-Blasius' Method Prandtl Flow past a Flat Plate-Direct Method References.

John M. Miller

Papers

Robust Computational Techniques for Boundary Layers