The Semi-Discrete Approach

Abstract

One strategy for obtaining finite-difference approximations to a PDE is to start by differencing the space derivatives only, without approximating the time derivative. In the following chapters, we proceed with an analysis making considerable use of this concept, which we refer to as the semi-discrete approach. Differencing the space derivatives converts the basic PDE into a set of coupled ODE’s. In the most general notation, these ODE’s would be expressed in the form

$$\frac{{d\vec u}}{{dt}} = \vec F(\vec u,t),$$

(4.1)

which includes all manner of nonlinear and time-dependent possibilities. On occasion, we use this form, but the rest of this chapter is devoted to a more specialized matrix notation described below.