The nonstationary convection-diffusion equation

Abstract

The equation to be studied in this chapter is the nonstationary convection diffusion equation:

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

(5.1)

See Sect. 2.3 for initial and boundary conditions and further comments, and Theorem 2.4.3 for the maximum principle satisfied by (5.1). Discretization of the spatial derivatives has been discussed in the preceding chapters. In the present chapter, attention will be focussed on temporal discretization. As a consequence, numerical stability will loom large. As in the preceding chapter, the aim is not merely to discuss the numerical solution of the relatively simple equation (5.1), but to prepare the ground for the numerical solution of the equations of fluid dynamics. This will guide our selection of numerical methods to be discussed, from the plethora of schemes available. Further introducation to the subject may be found in Morton (1996), Strikwerda (1989).