Gauss-Seidel point iteration method

Multigrid acceleration of the Gauss-Seidel point-iterative method is currently used in many commercial CFD codes to solve the system of algebraic equations resulting from the discretization of the governing equations. For this reason, the basic principles and nomenclature must be known by the users of commercial codes and in particular for researchers that are making their own codes. 

There are two basic families of solution techniques for linear algebraic equations Direct- and iterative methods. A well known example of direct methods is Gaussian elimination. The simultaneous storage of all coefficients of the set of equations in core memory is required. Iterative methods are based on the repeated application of a relatively simple algorithm leading to eventual convergence after a number of repetitions (iterations). Well known examples are the Jacobi and Gauss-Seidel point-by-point iteration methods. 

The differential equations describing heat transfer through the fabric, air gap and skin are solved by a finite difference model. Due to the non-linear terms of absorption of incident radiation, Gauss-Seidel point-by-point iterative scheme is used to solve these equations. An under-relaxation procedure is utilized to avoid divergence of the iteration method. The Crank-Nicolson scheme [49] is used to solve the resulting ordinary differential equations in time. 

This means that now a new value of,+,, T cannot be calculated only from values at time i as in Eq. (5.4-2) but that all the new values of T at t -(- At at all points must be calculated simultaneously. To do this an equation is written similar to Eq. (5.4-26) for each of the internal points. Each of these equations and the boundary equations are linear algebraic equations. These then can be solved simultaneously by the standard methods used, such as the Gauss-Seidel iteration technique, matrix inversion technique, and so on (G1,K1). 

---