Overrelaxation

If P = I, this is the Gauss-Seidel method. If > I, it is overrelaxation if P < I it is underrelaxation. The value of may be chosen empirically, 0 < P < 2, but it can be selected theoretically tor simple problems hke this (Refs. 106 and 221). In particular, these equations can be programmed in a spreadsheet and solved using the iteration feature, provided the boundaries are all rectangular. 

This is called point-simultaneous overrelaxation. If we set k = [s]nn, we have obtained the discrete formulation of Van Cittert s method. This connection between Van Cittert s method and the classic iterative methods of solving simultaneous equations was demonstrated in an earlier work (Jansson, 1968, 1970). 

For successive overrelaxation, we understand Eq. (26) to incorporate the use of o(k+1 values in place of o k) values in the convolution product as soon as they are formed for preceding x values. This adaptation can be explicitly displayed by the appropriate use of the Heaviside step function in a modified version of Eq. (26). The method of Van Cittert is a special case of simultaneous relaxation in which C = 1. 

There are many iterative methods (Jacobi, Gauss—Seidel, successive overrelaxations, conjugate gradients, conjugate directions, etc.) characterized by various choices of the matrix M. However, very often the most successful iterative processes result from physico-chemical considerations and, hence, corresponding subroutines cannot be found in normal computer libraries. 

Nicholls, A. andB. Honig. (1991). A rapid finite difference algorithm utilizing successive overrelaxation to solve the Poisson-Boltzmann equation. J. Comp. Chem. 12 435445. 

Both point-by-point and line-by-line overrelaxation methods were used to resolve the algebraic equations. ° An overrelaxation parameter of 1.5-1.8 was typically used. The two methods required similar computational times. An upwind scheme was used for all variables for high-Pe problems, while a central-difference scheme was used for low Pe. For some high-Pe cases, a central-difference scheme was used for the potential, but no appreciable differences in the results were observed. 

The successive overrelaxation method is a variant of the Gauss-Seidel method, wherein the ( H-l)th iteration is a weighted average of the Ganss Seidel h and ( -tl)th estimates x, and xf , respectively. The reader is referred to Jensen and Jeffreys (1977) for a detailed account on the matrices and solution methods. 

Similarly to the two-dimensional case, the convergence was improved by the method of successive overrelaxation (8). Between one-half to one-tenth of the original computer time for the successive displacement method was required to obtain an accuracy of 0.005%. Nevertheless, as... 

There are three commonly used iterative methods which we will briefly present here. They are Jacobi, Gauss-Seidel and the successive overrelaxation methods. 

In many problems, the iterated solutions approach the exact solutions in a monotonic fashion. Therefore, it is useful in this case to speed up the convergence process by overrelaxing the iterated solutions. The equation for the overrelaxation scheme is modified from the Gauss-Seidel equation... 

The proof of this last statement uses only the non-negative irreducible and convergent nature of the matrix M. In order to sharpen this last result, as well as introduce the basis for the successive overrelaxation iterative method of Young and Frankel [52 12], we make the following definition. 

The quantity o> in (4.11) is called the relaxation factor. We observe that, for CO = 1, this iterative method reduces to the Gauss-Seidel iterative method of (4.8)-(4.8 ). For reasons of brevity, we shall say that a matrix G, which is cyclic of index 2, is consistently ordered [52] if it is the form of (4.10). With the concept of a consistent ordering. Young [52] established the following general relationship between the eigenvalues A of the successive overrelaxation matrix... 

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