ABSTRACT
Partial differential equations are involved in the description of virtually every physical situation where quantities vary in space or in space and time. These include phenomena as diverse as diffusion, electromagnetic waves, hydrodynamics, and quantum mechanics (Schroedinger waves). In all but the simplest cases, these equations cannot be solved analytically and so numerical methods must be employed for quantitative results. In a typical numerical treatment, the dependent variables (such as tem perature or electrical potential) are described by their values at discrete points (a lattice) of the independent variables (e.g., space and time) and, by appropriate discretization, the partial differential equation is reduced to a large set of difference equations. Although these difference equations then can be solved, in principle, by the direct matrix methods discussed in Chapter 5, the large size of the matrices involved (dimension compa rable to the number of lattice points, often more than several thousand) makes such an approach impractical. Fortunately, the locality of the original equations (i.e., they involve only low-order derivatives of the de pendent variables) makes the resulting difference equations “sparse” in the sense that most of the elements of the matrices involved vanish. For such matrices, iterative methods of inversion and diagonalization can be very efficient. These methods are the subject of this and the following chapter.
