The stationary convection-diffusion equation

Abstract

The convection-diffusion equation has been derived in Sect. 1.11. We take ρ = 1, and obtain

$$ \frac{\partial\varphi}{\partial t} + u_{\alpha} \varphi,\alpha - (D\varphi,\alpha),\alpha = q, x \in \Omega \subset {\mathbb R}^d, 0 < t \leq T.$$

For the physical significanece of the terms in this equation, see Sect. 1.11. The equation is assumed to be linear, with φ the only unkown. A number of important aspects of the numerical analysis of the equations of fluid dynamics show up in this simple equation. Its simplicity allows a thorough analysis of these aspects, which will be given in this chapter. Readers who have some experience in computational fluid dynamics man think at first sight that our treatmen of such a simple linear equation is too detailed, but it is a fact that sometimes cotroversial and not always well understood important issues, notably the occurrence of numerical ‘wiggles’, the specification of outflow boundary coditions, singular perturbation aspects (the occurrence of boundary layers when D << 1, in a sense to be made precise shortly, a common situation in fluid dynamics and the role of false (numerical) viscosity, can be brought out and clarified completely in the context of this simple equation.