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 like 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’ll get warning about circular dependence) but becomes impractical for more than “toy” problems.