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Applied Computational Fluid Dynamics Techniques - Wiley Episode 1 Part 7 doc
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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
K · u = f. (7.1)
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)):
K · u = f → U · u = f
. (7.2)
To see how this is done in more detail, and to obtain an estimate of the work involved, we
rewrite (7.1) as
Kijuj = f i
. (7.3)
Suppose that the objective is to obtain vanishing entries for all matrix elements located in
the j th column below the diagonal Kjj entry. This can be achieved by adding to the kth row
(k>j ) an appropriate fraction of the j th row, resulting in
(Kkl + αkKjl)ul = f k + αkf j , k > j. (7.4)
Applied Computational Fluid Dynamics Techniques: An Introduction Based on Finite Element Methods, Second Edition.
Rainald Löhner © 2008 John Wiley & Sons, Ltd. ISBN: 978-0-470-51907-3
138 APPLIED COMPUTATIONAL FLUID DYNAMICS TECHNIQUES
=
K uf
L
0
0
0
=
U uf
0
0
0
=
K uf
L
0
0
0
=
L uf
0
0
0
t
=
K uf
=
0
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a)
K u f'
b)
c)
Figure 7.1. Direct solvers: (a) Gaussian elimination; (b) Crout decomposition; (c) Cholesky
Such an addition of rows will not change the final result for u and is therefore allowable. For
th