Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

Noting that
the above becomes
ˆ h ÿ H and
Elimination of HU i immediately yields
@h @
ˆ
@t @t
@ @
‡ …HU
@t @x
i†ˆ0 …7:22a†
i
@…HUi† @
‡ gH ˆ 0 …7:22b†
@t @xi @ 2
@
ÿ
@t2 @xi gH @
@x i
ˆ 0 …7:23†
or the standard Helmholtz wave equation. For this, many special solutions are
analysed in the next chapter.
The shallow-water equations derived in this section consider only the depthaveraged
¯ows and hence cannot reproduce certain phenomena that occur in
nature and in which some velocity variation with depth has to be allowed for. In
many such problems the basic assumption of a vertically hydrostatic pressure
distribution is still valid and a form of shallow-water behaviour can be assumed.
The extension of the formulation can be achieved by an a priori division of the ¯ow
into strata in each of which di€erent velocities occur. The ®nal set of discretized
equations consists then of several, coupled, two-dimensional approximations.
Alternatively, the same e€ect can be introduced by using several di€erent velocity
`trial functions' for the vertical distribution, as was suggested by Zienkiewicz and
Heinrich. 6 Such generalizations are useful but outside the scope of the present text.
7.3 Numerical approximation
Both ®nite di€erence and ®nite element procedures have for many years been used
widely in solving the shallow-water equations. The latter approximation has been
applied relatively recently and Kawahara 7 and Navon 8 survey the early applications
to coastal and oceanographic engineering. In most of these the standard procedures
of spatial discretization followed by suitable time-stepping schemes are adopted. 9ÿ16
In meteorology the ®rst application of the ®nite element method dates back to 1972,
as reported in the survey given in reference 17, and the range of applications has been
increasing steadily. 4;5;18ÿ41
At this stage the reader may well observe that with the exception of source terms,
the isothermal compressible ¯ow equations can be transformed into the depthintegrated
shallow-water equations with the variables being changed as follows:
…density† !h …depth†
u i …velocity† !U i …mean velocity†
p …pressure† ! 1
2 g…h2 ÿ H 2 †
Numerical approximation 223