Applied Computational Fluid Dynamics Techniques: An Introduction Based on Finite Element Methods, 2nd Edition

7 SOLUTION OF LARGE SYSTEMS OF EQUATIONS

As we saw from the preceeding sections, both the straightforward spatial discretization of a steady-state problem and the implicit time discretization of a transient problem will yield a large system of coupled equations of the form

There are two basic approaches to the solution of this problem:

• (a) directly, by some form of Gaussian elimination; or
• (b) iteratively.

We will consider both here, as any solver requires one of these two, if not both.

7.1. Direct solvers

The rapid increase in computer memory, and their suitability for shared-memory multiprocessing computing environments, has lead to a revival of direct solvers (see, e.g., Giles et al. (1985)). Three-dimensional problems once considered unmanageable due to their size are now being solved routinely by direct solvers (Wigton et al. (1985), Nguyen et al. (1990), Dutto et al. (1994), Luo et al. (1994c)). This section reviews the direct solvers most commonly used.

7.1.1. GAUSSIAN ELIMINATION

This is the classic direct solver. The idea is to add (subtract) appropriately scaled rows in the system of equations in order to arrive at an upper triangular matrix (see Figure 7.1(a)):

To see how this is done in more detail, and to obtain an estimate of the work involved, we rewrite (7.1 ...