Abstract
Often, the most time-consuming step in solving partial differential equations in two space partial differential equations in two space dimensions is an iterative solution of the finite-difference equations. For the closed boundary case, this solution is difficult when the equations are anisotropic, as they are normally when mesh spacing is much longer in one direction than in the other. This paper presents a solution method for use in such problems. The method is shown to be very fast in an anisotropic problem, but not as fast as other available methods in an isotropic problem. the extension to three space dimensions is outlined.
Introduction
In the finite-difference solution of multidimensional parabolic or elliptic equations, a set of many linear parabolic or elliptic equations, a set of many linear simultaneous algebraic equations arises. The most time-consuming part of the solution is the solving of this set of equations, normally accomplished by using some iterative method. Often the finite-difference equations are anisotropic. By anisotropic it is meant that two of the of diagonal coefficients in each equation are much larger than the other off-diagonal coefficients. When this occurs, the solution becomes more difficult for the closed boundary case. Until recently it appeared that line successive overrelaxation was the best technique to use in such a situation. Now, a method developed by Stone appears best among those published. This paper describes an alternative method for use with anisotropic systems. The method can be considered to be a specialization of a more general technique discussed by De la Vallee Poussin. It consists of a correction applied at each mesh point in a line, coupled with line successive overrelaxation. The method is shown to be very fast in a two-dimensional anisotropic problem. The extension to three dimensions is outlined, but no calculations are presented for that case. A theoretical analysis of the new method will be the subject of a later paper. paper.
METHOD
Consider the physical system from which the finite-difference equations are derived, as shown in Fig. 1. The system is represented by a matrix of mesh points. The finite-difference equation at each mesh point takes the form
.........(1)
All coefficients with the exception of di, j, are nonnegative. If the problem is elliptic (steady-slate), bi, j is zero. This equation is more commonly written as follows.
.....(2)
where
This equation is written for each mesh point to create the set of simultaneous algebraic equations that must be solved. The matrix formed from the coefficients of these equations is often symmetric. The mesh points in Fig. 1 are arranged in a square pattern.
SPEJ
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Finite-difference equations of quasistatic motion of the shallow concrete shells in nonlinear setting