Introduction

In the previous chapters we studied a variety of solution methods for a large number of PDEs. We point out, though, that the applicability of these methods is limited to canonical equations in simple domains. Equations with nonconstant coefficients, equations in complicated domains, and nonlinear equations cannot, in general, be solved analytically. Even when we can produce an ‘exact’ analytical solution, it is often in the form of an infinite series. Worse than that, the computation of each term in the series, although feasible in principle, might be tedious in practice, and, in addition, the series might converge very slowly. We shall therefore present in this chapter an entirely different approach to solving PDEs. The method is based on replacing the continuous variables by discrete variables. Thus the continuum problem represented by the PDE is transformed into a discrete problem in finitely many variables. Naturally we pay a price for this simplification: we can only obtain an approximation to the exact answer, and even this approximation is only obtained at the discrete values taken by the variables.

The discipline of numerical solution of PDEs is rather young. The first analysis (and, in fact, also the first formulation) of a discrete approach to a PDE was presented in 1929 by the German-American mathematicians Richard Courant (1888–1972), Kurt Otto Friedrichs (1901–1982), and Hans Lewy (1905–1988) for the special case of the wave equation.