Successive overrelaxation

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. 

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. 

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... 

Theorem The successive overrelaxation method with optimum relaxation factor converges at least twice as fast as the Chebyshev semi-iterative method with respect to the Jacobi method, and therefore at least twice as fast as any semi-iterative method with respect to the Jacobi method. Furthermore, as the number of iterations tends to infinity, the successive overrelaxation method becomes exactly twice as fast as the Chebyshev semi-iterative method. 

As in the case of successive overrelaxation, the efficiency of the application of Chebyshev polynomials in accelerating the outer iterations depends upon the accurate estimation of the particular constant, 5, the dominance ratio for the matrix T. A practical numerical method for estimating <7 is given in [45]. 

A comparison of the successive overrelaxation method and semi-iterative... 

Figure 6 Error reduction rate of the successive overrelaxation method. The smaller slope of the curve for small values of the relative error indicates the poor performance of solvers based on this method.

Lower-Upper Symmetric Successive Overrelaxation (LU-SSOR) Scheme... 

Equations (7.3.11) and (7.3.12) or (7.3.13) and (7.3.14) may be solved by introducing the concept of the vorticity and the stream function [15] the appropriate finite difference form of vorticity transport and stream equations may then be solved by successive overrelaxation. 

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