Numerical Math - The Gauss-Seidel (GS) method

The Gauss-Seidel method is an improvement upon the basic Jacobi method [1]. The GS method utilizes known values and current updates them in the algorithm as opposed to the Jacobi method which sweeps across the domain without utilizing known values to accelerate convergence. The accelerated convergence also produces less round-off error and utilizes less memory since only one generation of guesses is needed.

If the sweep begins at the top left corner then the GS algorithm looks like

\begin{equation} U_{i, j} = \dfrac{1}{4} \left[ U^{(old)}_{i + 1, j} + U^{(new)}_{i - 1, j} + U^{(old)}_{i, j + 1} + U^{(new)}_{i, j - 1} \right] \end{equation}

Compare to the Jacobi method.  Both algorithms shown are for the square wire finite difference technique problem in Landau et al [1].

\begin{equation} U_{i, j} = \dfrac{1}{4} \left( U_{i + 1, j} + U_{i - 1, j} + U_{i, j + 1} + U_{i, j - 1} \right) \end{equation}

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References:

[1] R. H. Landau, M. J. Páez, and C. C. Bordeianu. A Survey of Computational Physics - Introductory Computational Science, Princeton University Press, Princeton, New Jersey. 2008