Environmental Engineering Reference
In-Depth Information
finite differences; this only modifies the advection term and decenters it in the
direction of flow or in the opposite direction. It thus appears that the upstream
decentering (upwind) is favorable, and a qualitatively correct solution can be
obtained for any Péclet number. Quantitatively, however, there are still mistakes.
The modification is therefore interesting, but not perfect.
To implement this idea in the finite element method, we must change the
weighting functions [ZIE 89]. In Galerkin's method, the weighting functions are
identical to the interpolation functions. The S
tandard upwind Petrov-Galerkin
method
can overcome this limitation and generalize upstream decentering. In a one-
dimensional state, it takes weighting functions with the following expression:
WN
α
=
+
[13.49]
L
L
L
and various forms of correction may be adopted (see Figure 13.5). The easiest way
is to choose:
()
∂
N
ξ
ξ
W
=
[13.50]
L
∂
but higher-order functions have also been proposed.
~
N
L
W
L
1
1
x
x
h
h
W
L
α
x
h
Figure 13.5.
Corrections of the one-dimensional Petrov Galerkin
weighting function method
If the analytical solution of the one-dimensional problem is known, it is possible
to obtain coefficient α by adjusting it so as to have a good correspondence between
the numerical and analytical solutions (Figure 13.4, Petrov-Galerkin α
opt
solution). It
then varies according to the ratio between the advective and diffusive effects,
measured by the dimensionless Péclet number:
