346
7. Physically Insignificant Fast Waves
dispersion relation associated with wave-like solutions to more complex systems
of goveming equations (such as the shallow-water system discussed in the next
section) often has individual roots of the form
and (7.23) serves as the simplest differential equation describing the time dependence of such waves.
The simplest semi-implicit approximation to (7.23) is
ifJn+1 ifJn
------'-- + illJHifJn+1 + iWLifJ n = O.
/1t
The stability and the accuracy of this scheme have already been analyzed in connection with (2.33); it is first-order accurate and is stable whenever IWLI < IWHI.
Since IWLI < IWHI by assumption, the method is stable for all /1t. The weakness
of this scheme is its low accuracy. A more accurate second-order method can be
obtained using the centered-in-time formula
(ifJ n+1 + ifJn-l) . n
ifJn+l - ifJn-l
.
-'---2-/1-'t-- + IWH
2
+ IWLifJ = o.
(7.24)
In order to investigate the stability of this method, consider the behavior of oscillatory solutions of the form exp( -iwn/1t) , which satisfy (7.24) when
(7.25)
where
Defining tan ß = WH, (7.25) becomes
sin W = tan ßcos W+ WL,
or equivalently,
sinwcosß - sinß cos ö =
By the Pythagorean theorem, cos ß = (1
sin(w - ß) = WL(1 +
cosß.
and the preceding reduces to
or equivalently,
W =
+ arcsin (wL(1 +
.
The semi-implicit scheme (7.24) will be stable when the Wsatisfying this equation
are real and distinct, which is guaranteed when