Page 181 - Computational Fluid Dynamics for Engineers

168 5. Numerical Methods for Model Hyperbolic Equations

This equation gives the stability requirement for the solution of the difference
equation, Eq. (4.4.3a), by placing a constraint on the size of the time step
relative to the size of the mesh spacing.
The von Neumann or Fourier stability method can also be applied to examine
the stability characteristics of hyperbolic equations. To illustrate this, consider
the linear convection equation, (5.1.1),

and replace the time derivative with a forward difference, Eq. (4.3.8), and the
spatial derivative with a central difference. Eq. (4.3.7): this gives the simple
Euler explicit scheme

~
+ c l+ \ A l l = 0 (5.7.17)
At 2Ax
which can be shown to be unstable by applying the von Neumann stability
analysis (see Problem 5.7). If in the time derivative, the term uf in Eq. (4.7.17)
is replaced by an average value between grid points (i -f 1) and (i — 1), so that
Eq. (5.7.17) becomes

At + C 2Ax _ 0 ( 5 7 1 8 j
- '
the scheme becomes stable by \cAt/Ax\ < 1. This differencing in the time
derivative is called the Lax Method.
To apply the von Neumann stability analysis to Eq. (5.7.18), again assume
an error in the form given by Eq. (5.7.7) and substitute this form into the
difference equation, (5.7.18); the amplification factor becomes

e aAt = cos(k mAx) - io sin(fc m Az) (5.7.19a)
and in terms of phase angle 0, it can be written as

2
2
G = |G| e^ = ^cos (kmAx) + a 2 sin (fc m Ar) e^ (5.7.19b)
where
t
(j) = a n - 1 [—crtan(fc mZ\x)]
aAt
The stability requirement \e \ < 1, when applied to Eq. (5.7.19) yields

a = cf x < 1 (5.7.20)
where a = cAt/Ax is called the Courant number. Eq. (5.7.20) is known as the
Courant-Friedrichs-Lewy condition, generally referred to as the CFL condition.
Figure 5.4 shows a polar plot of Eq. (5.7.19b) for several Courant numbers.
For a = 1 all frequency components are propagated without attenuation in the