Splitting and alternating direction 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.

Alternating Direction Implicit Methods

Finite difference methods for linear parabolic equations

Heat transport at the microscale is of vital importance in microtechnology applications. In this study, we develop a finite difference scheme of the Crank-Nicholson type by introducing an intermediate function for the heat transport equation at the microscale. It is shown by the discrete energy method that the scheme is unconditionally stable. Numerical results show that the solution is accurate. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 697–708, 1999

A finite difference scheme for solving the heat transport equation at the microscale

Unconditionally Stable Finite Difference Scheme and Iterative Solution of 2D Microscale Heat Transport Equation

In previous papers [12], [13], [14] the author developed a difference analogue of the energy method for determining the stability of difference approximations to partial differential equations with variable coefficients. The purpose of this paper is to apply this method to establish the unconditional stability of two types of difference approximations to parabolic differential equations, the (implicit) alternating direction methods of DOUGLAS, PEACEMAN, and RACHFORD [3], [5], [15], and a new semi-explicit method. For the model problem, the first boundary value problem for the heat conduction equation in a rectangular domain, the unconditional stability of the alternating direction methods was proved in [3] and [3]. The proof consists in showing, with the aid of Fourier analysis, that the yon Neumann stability condition [4], [11] is always satisfied. It can be shown [1], however, that this method of proof cannot be extended beyond the model problem. With the aid of the energy method we prove that the results in E3] and [6] can be extended beyond the model problem. We first treat, as a typical case, the heat conduction equation in a cylindrical domain with an essentially arbitrary, bounded base. Then we indicate briefly the extension to parabolic equations with variable coefficients. The second type of difference method we term the semi-explicit method, because it is an explicit method only for certain orderings of the net points. The idea for this difference method comes from the observation that there is a formal correspondence between parabolic difference equations and iterative methods for solving elliptic difference equations; the semi-explicit method corresponds to the well known method of successive displacements [7]. The only other known example of an unconditionally stable explicit difference method is due to Do FORT and FRANKEL [6]. Their method, however, requires two lines of initial data to start the solution, while the semi-explicit method is self-starting. On the other hand, both of these methods involve a similar local truncation error, and they must be subjected to a mild mesh ratio condition in order to be consistent [11] with the differential equation being approximated. For other applications of the energy method to the stability problem for partial difference equations, see FRIEDRICHS [8], KREISS [9], LAX [10] and LEES

Alternating Direction and Semi-Explicit Difference Methods for Parabolic Partial Differential Equations

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Alternating Direction and Semi-Explicit Difference Methods for Parabolic Partial Differential Equations