Numerical Methods and Its Solution - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 195

The matrix containing the non-zero coefficients is therefore manipulated into:
⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

⎡
⎣

⎤
⎦

B 1
B 2
···
B i
···
B n − 1
B n

A 12

1 0 0 0 0 0 φ 1

φ 1
···
φ i
···
φ n − 1
φ n

01 A 23 0 0 0 0

0 ··· ··· ··· 0 0 0

00 0 1 A ii + 1 0 0 =

00 0 ··· ··· ··· 0

00 0 0 0 1 A n − 1 n

0 0 0 0 0 0 1

The second stage involves back substitution , to evaluate φ nm and φ i as

φ n = B n φ i = B i − φ i + 1 A ii + 1 and (7.52)

The TDMA is more economical than the Gaussian elimination because of the absence
of arithmetic operations (multiplication and divisions) in obtaining φ i during back
substitution.
As mentioned earlier during the forward elimination stage the term A nn may
consist of a zero value. The reader should be aware that in order to prevent ill-
conditioning of a matrix for both of the direct methods presented, it is necessary to
ensure that

| A ii | > | A ii − 1 | + | A ii + 1 | (7.53)

This means that the diagonal coefficients are required to be much larger than the
sum of the neighbouring coefficients. If this is not the case, matrix row swapping
(otherwise known as pivoting ) can be performed.

7.3.2 Iterative Methods

Direct method such as Gaussian elimination can be employed to solve any system of
equations. Unfortunately, in most CFD problems that usually result in a large system
of non-linear equations, the cost of using this method is generally quite high as it
requires a lot pre-conditioning of the matrix to prepare it for the direct method process.
This leaves the option of employing iterative methods. In an iterative method, one
guesses the solution, and uses the equation to systematically improve the solution
until it reaches a specified level of convergence. If the number of iterations is small in
achieving convergence, an iterative solver may cost less to use than a direct method.
This is usually the case for CFD problems.
The simplest method from the various classes of iterative methods is the Jacobi
method which solves the left hand side of the matrix expression, using previous
values for φ on the right hand side. Let us revisit the system of equations, A φ = B ,