6.2. Upstream Weighted Methods
159
FIGURE 6.3. A standard element -1 ::; ~ ::; 1 in a loeal eoordinate system. (a) basis
funetion; (b) weighting funetion; (e) modifying funetion.
linear basis functions corresponding to the ends - 1 and 1 are
{
(6.2.11)
The weighting functions for the UWFEM or the Petrov-Galerkin method are
given as
where
{
Wl(~) = W2(~) = F(~) = i(l - ~)(1 + ~)
(6.2.12)
(6.2.13)
is called the modifying function and C( is a parameter to be determined. The
basis functions, weighting functions, and modifying functions are shown in
Figure 6.3. In order to illustrate the upstream weighted effect of this method,
let us consider the following steady one-dimensional advection-dispersion
equation:
oe
a 2 e
v ax - D ax2 = O.
(6.2.14)
In this case, Rij and F i in Eqs. (5.1.13) to (5.1.15) are equal to zero, and Aij is
reduced to
r ( aw a Aij = J(R) D ax' 0; + vw; 0; dx.
(6.2.15)
To derive the discrete finite element equation for node i, only Ai,i-l' Ai,i and
Ai,i+l' need to be calculated. According to the selection ofthe basis functions,
the integral of Eq. (6.2.15) is non-zero only in the subregion contains both
nodes i andj. Thus, for A i,i+1' we have
f
Xi +
1 (
aw 8 a Ai,i+l = Xi
D ax' 0; + vw; 0; dx,
(6.2.16)
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