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6. Numerical Solutions of Advection-Dominated Problems
Substituting Eq. (6.1.3) into Eq. (6.1.1), we have
ßn + VO"n - iDO"; = 0
or
ßn = O"n(iDO"n - V).
(6.1.4)
Thus, the relationship between frequency ßn' and wave number O"n is established. Substituting Eq. (6.1.4) into Eq. (6.1.3), we obtain
C ~ C n exp [iO"n(x - Vt)] exp( - DO";t),
where the first exponent on the right-hand side represents the displacement
of the nth wave and the second exponent represents the variation of wave
amplitude. After a time increment 11t, the wave moves a distance V 11t, while
its amplitude changes with the factor exp( - DO";~t). Since
we have
C tHt ~ C n exp[ißn(t + ~t)] exp(iO"nx)
= Cn exp(ißnt + iO"nx) exp(ißn l1t),
(6.1.5)
In Eq. (6.1.5), exp(ißnl1t) is called the analytic enlargement factor and written
as An" From Eq. (6.1.4), we know
An = exp(ißnl1t) = exp( - DO"; 11t) exp( - iO"n V ~t),
thus the magnitude of the analytic enlargement factor is
IAnl = exp( -DO";~t),
and its phase angle is
On = O"n V~t.
(6.1.6)
(6.1.7)
Similarly, we can define the numerical enlargement factor of the nth wave,
A~, which shows the ratio between the numerical solutions of time t + ~t and
t. Once we know the discrete scheme of a numerical solution, it is easy to find
the expression for A~. For example, the implicit scheme of finite difference of
Eq. (6.1.1) is (see Eq. (4.1.14)):
[ DI1t
VI1tJ
[2D~tJ
[DI1t
VI1tJ
- (~X)2 + 2~x Ci-l,k+l + 1 + (~X)2 Ci,k+l - (~xf - 2~x Ci+l,k+l
= Ci,k
(6.1.8)
U sing the dimensionless parameters
-
D~t
D = (~X)2'