Alternating direction and semi-explicit difference methods for parabolic partial differential equations

By M. Lees. 2018

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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

Subfields: Applied Mathematics, Numerical Analysis

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