Finite differences

The opening line of Anna Karenina, ‘All happy families resemble one another, but each unhappy family is unhappy in its own way’, is a useful metaphor for the computation of ordinary differential equations (ODEs) as compared with that of partial differential equations (PDEs). Ordinary differential equations are a happy family; perhaps they do not resemble each other but, at the very least, we can write them in a single overarching form y′ = f(t, y) and treat them by a relatively small compendium of computational techniques. (True, upon closer examination, even ODEs are not all the same: their classification into stiff and non-stiff is the most obvious example. How many happy families will survive the deconstructing attentions of a mathematician?)

Partial differential equations, however, are a huge and motley collection of problems, each unhappy in its own way. Most students of mathematics will be aware of the classification into elliptic, parabolic and hyperbolic equations, but this is only the first step in a long journey. As soon as nonlinear – or even quasilinear – PDEs are admitted for consideration, the subject is replete with an enormous number of different problems and each problem clamours for its own brand of numerics. No textbook can (or should) cover this enormous menagerie. Fortunately, however, it is possible to distil a small number of tools that allow for a well-informed numerical treatment of several important equations and form a sound basis for the understanding of the subject as a whole.