Alternating Direction Implicit Methods

Publisher Summary Alternating direction implicit methods, or ADI methods as they are called for short, constitute powerful techniques for solving elliptic and parabolic partial difference equations. However, in contrast with systematic overrelaxation methods, their effectiveness is hard to explain rigorously with any generality. Indeed, to provide a rational explanation for their effectiveness must be regarded as a major unsolved problem of linear numerical analysis. The current status of this problem with regard to the elliptic partial difference equation in the plane is discussed in the chapter. It is divided into four chapters and four appendices. Part I deals with ADI methods that iterate a single cycle of alternating directions. In this case, the theory of convergence is reasonably satisfactory. Part II studies the rate of convergence of AD1 methods using m > 1 iteration parameters, in the special case that the basic linear operators H, V, Σ in question are all permutable. In this case, the theory of convergence and of the selection of good iteration parameters is now also satisfactory. Part III surveys that that is known about the comparative effectiveness of ADI methods and methods of systematic overrelaxation, from a theoretical standpoint. Part IV analyzes the results of some systematic numerical experiments that were performed to test comparative convergence rates of different methods. The four appendices deal with various technical questions and generalizations.