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

36 Convection dominated problems
and for a typical nodal point i, wehave
n ‡ 1
xi ˆ x n i ‡
… tn‡1
t n
U dt …2:67†
where in general the `velocity' U may be dependent on x. However, if F ˆ F… † and
U ˆ @F=@ ˆ U… † then the wave velocity is constant along a characteristic by virtue
of Eq. (2.65) and the characteristics are straight lines.
For such a constant U we have simply
n ‡ 1
xi ˆ x n i ‡ U t …2:68†
for the updated mesh position. This is not always the case and updating generally has
to be done with variable U.
On the updated mesh only the time-dependent di€usion problem needs to be
solved, using the methods of Volume 1. These we need not discuss in detail here.
The process of continuously updating the mesh and solving the di€usion problem
on the new mesh is, of course, impracticable. When applied to two- or three-dimensional
con®gurations very distorted elements would result and di culties will always
arise on the boundaries of the domain. For that reason it seems obvious that after
completion of a single step a return to the original mesh should be made by interpolating
from the updated values, to the original mesh positions.
This procedure can of course be reversed and characteristic origins traced backwards,
as shown in Fig. 2.11(b) using appropriate interpolated starting values.
The method described is somewhat intuitive but has been used with success by
Adey and Brebbia 45 and others as early as 1974 for solution of transport equations.
The procedure can be formalized and presented more generally and gives the basis of
so-called characteristic±Galerkin methods. 46
The di€usion part of the computation is carried out either on the original or on the
®nal mesh, each representing a certain approximation. Intuitively we imagine in the
updating scheme that the operator is split with the di€usion changes occurring
separately from those of convection. This idea is explained in the procedures of the
next section.
2.6.2 Characteristic±Galerkin procedures
We shall consider that the equation of convective di€usion in its one-dimensional
form (2.61) is split into two parts such that
ˆ ‡ …2:69†
and
@
@t
is a purely convective system while
@
@t
ÿ @
@x
@
‡ U ˆ 0 …2:70a†
@x
k @
@x
‡ Q ˆ 0 …2:70b†