Computational Fluid Structure Interaction - Computational Hemodynamics: Theory, Modelling and Applications - page 141

5.5.3 Solution for a One-Dimensional Steady Diffusion Equation

TDMA Method The solution for a system of equations for a 4-cell discretisation
in matrix form is given as

3000 − 1000 0 0 T

T
T
T

203000
3000
3000
803000

1

− 1000 2000 − 1000 0

2

=

0 − 1000 2000 − 1000

3

0 0 − 1000 3000

4

We first normalise the top row by making A 11 to unity (divide Row-1 by 3000). We
then multiply the first row by A 21 and subtract it from the second row.

1 − 1 / 3 0 0 T

T
T
T

67

2 3

1

0 1666 1000 0 70666

2

2 3

3 2

=

0 − 1000 2000 − 1000 3000

3

0 0 − 1000 3000 803000

4

Repeating the first step, but for the second row, A 22 is set to unity (divide Row-2
by 1666 3 ). We then multiply second row by A 32 and subtract it from the third row.

1 − 1/3 0 0 T

T
T
T

67

2

1 3

0 1 − 3/5 0

42

2

2 5

=

0 0 1400 − 1000

45400

3

0 0 − 1000 3000

803000

4

Repeating for the remaining rows of the matrix, we get:

1 − 1/3 0 0 T

T
T
T

67

2

1 3

0 1 − 3/5 0 42

2

2 5

=

0 0 1400 − 1000 45400

3

0 0 − 1000 3000 803000

4

The second stage of the TDMA simply involves back substitution . Using Eqn. (5.79),
the solution to the above system is

T = 140 5 .; T 218 5 .; T == 293 5 .; T 365 5 . =

1 2 3 4