Page 120 - Computational Fluid Dynamics for Engineers

106 4. Numerical Methods for Model Parabolic and Elliptic Equations

and dT/dt is expressed by the central-difference formula of Eq. (4.3.7),

dT .71+1/2 rjin+l _ rpn (4.4.10b)
Xi)
y
dt " («" ' ~ " At
Before considering the solution of Eq. (4.2.4) using this finite-difference ap-
proximation, note that a more general finite-difference approximation to this
equation is
n+\
rnTl+l rpn
= a + (1-6) 2 (4.4.11)
At dx2 dx
, ) t
where in practice O < 0 < 1 . 0 = O gives the explicit scheme, 6 — 1/2 the Crank-
Nicolson method, and S = l a fully implicit backward time-difference method.
The time differencing in Eq. (4.4.11) is known as the trapezoidal differencing
scheme as well as Crank-Nicolson differencing. The equations are uncondition-
ally stable and convergent for 1/2 < 6 < 1, but for 0 < 6 < 1/2

aAt
< (4.4.12)
2
Ax ~ 2(1 - 26)
For 9 = 1/2, Eq. (4.2.4) can be written as
+1
_ ^ £_L_ [(7 -+i - 27f + T-V) - 22? + 7?_i)]
1 7 ? + 1 = 2 + (T? +1
(
At 2 Ax '
or as
Ki< I -1 (4.4.13)
where
ai = l, bi — -(2 + X), Ci = \

-XT?-(T? +l-2T? + T?_ 1), l < i < I - l ,
(4.4.14)
2 Ax 2
A =
a At
n
Note that r^ contains only values of T , not T n+1 . At i — 0 and I, the boundary
conditions T 0 n+1 and T? +1 are given by Eq. (4.4.1). As a result, for i = 1 and
i = I — 1, Eq. (4.4.13) can be written as
nn+l n + i -n+1 1
1
n +
b xT^ + ciT 2 = n - aiT 0 = rf for i = 1 (4.4.15a)
and
_
+
i
a / - i 7 7 + 2 + 6 / _ i 3 7 + / = r / _ i - c / _ i T 7 •n+1 1 = r J _ 1 for i = / - l (4.4.15b)
With the boundary conditions given by Eq. (4.4.4), again use Eqs. (4.4.5) and
(4.4.8). For i = 0, it follows from Eq. (4.4.5) that