Numerical Methods and Its Solution - Computational Fluid and Particle Dynamics in the Human Respiratory System - page 196

as described in the previous section; the general form of the algebraic equation for
each unknown nodal variables of φ can be written as:

i − 1 n

A ij φ j + A ii φ i +

=

A ij φ j B i (7.54)

j = 1 j = i + 1

In Eq. (7.54), the Jacobi method requires that the nodal variables φ j (non-diagonal
matrix elements) are assumed to be known at iteration step k and the nodal variables
φ i are treated as unknown at iteration step k + 1. Solving for φ i ,wehave

⎛ ⎞

i − 1 n

1
A ii

φ (k + 1 )
i

⎝ B i −

A ij φ (k)
j

A ij φ (k)
j

⎠

= − (7.55)

j = 1 j = i + 1

The iteration process begins by an initial guess of the nodal variables φ j ( k = 0). After
repeated application of Eq. (7.55) to all the n unknowns, the first iteration, k =1, is
completed. We proceed to the next iteration step, k = 2, by substituting the iterated
values at k = 1 into Eq. (7.54) to estimate the new values at the next iteration step.
This process is continuously repeated for as many iterations as required to converge
to the desired solution.
A more immediate improvement to the Jacobi method is provided by the Gauss-
Siedel method in which the updated nodal variables φ ( k + 1 j are immediately used on
the right-hand side of Eq. (7.54) as soon as they are available. In such a case, the
previous values of φ ( k j that appear in the second term of the right-hand side of Eq.
(7.55) are replaced by the current values of φ ( k )
j

, which the equivalent of Eq. (7.55)

becomes

i − 1 n

B i
A ii −

A ij A ij

φ ( k + 1)
i

A ii φ ( k + 1) A ii φ ( k )

= − (7.56)

j j

j = 1 j = i + 1

Comparing the above two iterative procedures, the Gauss-Siedel iteration is typically
twice as fast as the Jacobi iteration. After repeated applications of Eqs. (7.55) and
(7.56), convergence can be gauged in a number of ways. One convenient condition
to terminate the iteration process is to ensure that the maximum difference between
each iteration, φ ( k + 1)
j

φ ( k j falls below some predetermined value. If the relative
change is continually increasing with each iteration, then the solution is diverging.

−

7.3.3 Solution for a One-dimensional Steady Diffusion Equation

TDMA method The solution for the steady heat conduction problem in a large brick
plate with a uniform heat generation in Sect. 7.2.3 is presented here. The system of