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Iteration Demonstrations

UPDATED: 09/15/2022

This workbook includes three separate demonstrations of Gauss-Seidel (Liebmann) iteration for the solution of well-ordered, sparse systems of linear equations. The first one, shown in the figure below, demonstrates using G-S to solve the system of linear equations arising from the finite-difference discretization of Laplace’s equation in 2-D.  You might enjoy comparing the current implementation with its predecessor from 35 years ago as seen in this video.

Another worksheet shows application of the Scarborough criterion to a set of two linear equations. The third worksheet shows the application of G-S in one-dimension and highlights the difficulty of applying pointwise iterative methods to large systems. The first and third demonstrations are animated.

Watching how these point-wise iterative methods perform gives strong motivation for studying more advanced methods.  These include, but are not limited to, successive-over-relaxation (SOR), the multigrid method, conjugate gradient methods and the modified strongly implicit method.   The latter operates behind the scenes in our two-dimensional, steady-state module.    Iteration can be done easily within an Excel spreadsheet.  You will get warnings about “circular dependence.”  Pointwise iteration as demonstrated here becomes impractical for more than “toy” problems.


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