Page 179 - Computational Fluid Dynamics for Engineers

166 5. Numerical Methods for Model Hyperbolic Equations

_1_ ^ n + l _ fn^ + j _ ( £n+l _ £n) = _ ^ _ {fn +i _ fn + fn^
2

+^(e2+ 1-2e? + et l) (5.7.2)

Since the exact solution T™ satisfies Eq. (4.4.3a), then the equation

_L ( £ n+l _ «) = _ ^ _ {£?+i _ 2£n + £ U ) ( 5 J . 3 )
£
represents the errors e™ and is identical to the basic scheme. Thus, the errors ef
1
evolve over time in the same manner as the numerical solution T/ . If the errors
Si decrease or stay the same, at some stage of the solution, the solution will be
stable as the solution progresses from time step n to n + 1; on the other hand,
if E^S become larger (amplify) during the progression from n to n + 1, then the
solution is unstable. Thus, for a stable solution,
£ n+l
< 1 (5.7.4)

In the stability analysis of a linear equation by the Fourier series method,
the error can be written in the form
ai ikmX
e{x,t)=^e e (5.7.5)
7 7 2 = 1
where a is a constant and k m is the wave number given by
777/7T
k
™ = -j- m = l , 2 , 3 , . . . (5.7.6)
with L corresponding to the length of the domain on which the equation is
being solved.
Since the difference equation under consideration is linear, and separate so-
lutions are additive (method of superposition), it is necessary to consider only
the propagation of the error due to one term of the series,

at [k
£m(x,t) = e e ™ x (5.7.7)
Substituting Eq. (5.7.7) into Eq. (5.7.3), we obtain
at
ea(t+At) eik mx _ at e\kmX _ ^ [ Cat cikm(x+Ax) _ ^ e e^ mx , eat eikm(x—Axh
e
(Ax) 2[
(5.7.8)
at lkrnX
If we divide by e e and use the relation
Ax
ik
[k
cos(k mAx) = -(e ™ Ax + e~ ™ )
then, Eq. (5.7.8) becomes