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

φ(x); t = 0 φ(x – Ut)
Fig. 2.10 The wave nature of a solution with no conduction. Constant wave velocity U.
and equations of this type can be readily discretized with self-adjoint spatial operators
and solved by procedures developed previously in Volume 1.
The coordinate system of Eq. (2.62) describes characteristic directions and the
moving nature of the coordinates must be noted. A further corollary of the coordinate
change is that with no conduction or heat generation terms, i.e. when k ˆ 0 and
Q ˆ 0, we have simply
@
ˆ 0
@t
or (2.65)
…x 0 †ˆ …x ÿ Ut† ˆconstant
along a characteristic [assuming U to be constant, which will be the case if F ˆ F… †].
This is a typical equation of a wave propagating with a velocity U in the x direction,
as shown in Fig. 2.10. The wave nature is evident in the problem even if the conduction
(di€usion) is not zero, and in this case we shall have solutions showing a wave
that attenuates with the distance travelled.
2.5.2 Possible discretization procedures
Transients ± introductory remarks 33
In Part I of this chapter we have concentrated on the essential procedures applicable
directly to a steady-state set of equations. These procedures started o€ from somewhat
heuristic considerations. The Petrov±Galerkin method was perhaps the most
rational but even here the amount and the nature of the weighting functions were a
matter of guess-work which was subsequently justi®ed by consideration of the numerical
error at nodal points. The Galerkin least square (GLS) method in the same way
provided no absolute necessity for improving the answers though of course the least
square method would tend to increase the symmetry of the equations and thus could
be proved useful. It was only by results which turned out to be remarkably similar to
those obtained by the Petrov±Galerkin methods that we have deemed this method to
be a success. The same remark could be directed at the ®nite increment calculus (FIC)
method and indeed to other methods suggested dealing with the problems of steadystate
equations.
For the transient solutions the obvious ®rst approach would be to try again the
same types of methods used in steady-state calculations and indeed much literature
x