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

Pe = Uh/2k = 0
= =
(All exact)
Pe = 1.0
Pe = 2.5
=
(Exact)
Pe = ∞
Standard Galerkin α = 0
Petrov–Galerkin α = 1.0 (full upwind difference)
Petrov–Galerkin α = α opt
Exact
h
Motivated by the fact that the propagation of information is in the direction of
velocity U, the ®nite di€erence practitioners were the ®rst to overcome the bad
approximation problem by using one-sided ®nite di€erences for approximating the
®rst derivative. 2ÿ5 Thus in place of Eq. (2.17a) and with positive U, the approximation
was put as
d
dx
L
Exact
~ i ÿ ~ i ÿ 1
h
The steady-state problem in one dimension 17
Fig. 2.2 Approximations to U d =dx ÿ k d 2 =dx 2 ˆ 0 for ˆ 0, x ˆ 0 and ˆ 1, x ˆ L for various Peclet
numbers.
1.0
…2:19†