ABSTRACT
This chapter is the extension from one to two dimensional steady state space models. The solution of the discrete versions of these can be approximated by various iterative methods, and here the successive over-relaxation and conjugate gradient methods will be implemented. Three application areas are diffusion in two directions, ideal and porous fluid flows in two directions, and the deformation of the steady state membrane problem. The model for the membrane problem requires the shape of the membrane to minimize the potential energy, and this serves to motivate the formulation of the conjugate gradient method. The classical iterative methods are described in G. D. Smith [23] and Burden and Faires [4]
Models of heat flow in more than one direction will generate large and nontridiagonal matrices. Alternatives to the full version of Gaussian elimination, which requires large storage and number of operations, are the iterative methods. These usually require less storage, but the number of iterations needed to approximate the solution can vary with the tolerance parameter of the particular method. In this section we present the most elementary iterative methods: Jacobi, Gauss-Seidel and successive over-relaxation (SOR). These methods are useful for sparse (many zero components) matrices where the nonzero patterns are very systematic. Other iterative methods such as the preconditioned conjugate gradient (PCG) or generalized minimum residual (GMRES) are particularly useful, and we will discuss these later in this chapter and in Chapter
Consider the cooling fin problem from the previous chapter, but here we will use the iterative methods to solve the algebraic system. Also we will study the effects of varying the parameters of the fin such as thickness, T , and width, W . In place of solving the algebraic problem by the tridiagonal algorithm as in Section 2.3, the solution will be found iteratively. Since we are considering a model with diffusion in one direction, the coefficient matrix will be tridiagonal. So, the preferred method is the tridiagonal algorithm. Here the purpose of using iterative methods is to simply introduce them so that their application to models with more than one direction can be solved.
