Overrelaxation method

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. 

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

In terms of actual arithmetic operations, the direct inversion of large submatrices combined with an iterative method tends to increase the amount of arithmetic operations per mesh point. This is, of course, to be balanced by an increase in the rate of convergence. In the case of the iterative method SLOR, either in two or three space dimensions, it fortunately can be shown [6] by suitable normalization of equations, that no additional arithmetic operations are required for the successive line overrelaxation method (SLOR) over what is required by the successive point overrelaxation method, while an improvement in rate of convergence is always obtained. 

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. 

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.

Budko, N. V., Samokhin, A. B., and Samokhin, A. A. [2005] A generalized overrelaxation method for solving singular volume integral equations in low-frequency scattering problems, Diff. Equat, 41,1262-1266. 

When w = 1, this method is exactly the same as the unmodified Gauss-Seidel. Methods for estimating the optimal w are given by Lapidus and Finder [4J, who also show that the overrelaxation method is five- to one-hundred times faster (depending on step size and convergence criterion) than the Gauss-Seidel method. 

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