Page 59 - Introduction to Computational Fluid Dynamics

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2.8 Methods of Solution 1D HEAT CONDUCTION
When coefficients AE i , AW i , and AP i are calculated and Su i and Sp i are suitably
updated to account for the effects of source linearization, boundary conditions, and
underrelaxation, we are ready to solve the set of equations (2.43) at an iteration
level l + 1. There are two extensively used methods for solving such equations.
2.8.1 Gauss–Seidel Method

The Gauss–Seidel (GS) method is extremely simple to implement on a computer.
The main steps are as follows:

1. At a given iteration level l, calculate coefficients AE, AW, AP, Su, and Sp
l
using temperature T for i = 2to N − 1
2. Hence, execute a DO loop:

100 FCMX = 0
DO1I=2,N-1
TL = T(I)
ANUM = AE(I) T(I+1) + AW(I) T(I-1) + SU(I)
∗
∗
ADEN = AE(I) + AW(I) + SP(I)
T(I) = ANUM / ADEN
FC = (T(I) - TL) / TL
IF (ABS(FC).GT.FCMX) FCMX = ABS(FC)
1 CONTINUE

3. If FCMX > CC, go to step 1.

The method is also called a point-by-point method because each node i is visited
in succession. The method is very reliable but requires a large number of iterations
and hence considerable computer time, particularly when N is large.

2.8.2 Tridiagonal Matrix Algorithm

In the tridiagonal matrix algorithm (TDMA), Equation 2.43 is rewritten as
T i = a i T i+1 + b i T i−1 + c i , (2.69)

where

AE i AW i Su i
a i = , b i = , c i = . (2.70)
AP i + Sp i AP i + Sp i AP i + Sp i
Note that since Sp i ≥ 0, a i and b i can only be fractions. Equation 2.69 represents
(N − 2) simultaneous algebraic equations. In matrix form, these equations can be
written as [A] [T] = [C], where the coefficient matrix [A] will appear as shown