6.2. Upstream Weighted Methods
167
sides to form a polygon surrounding node i, which is called the exclusive
subdomain for node i. Then, the equation of local mass balance over the
subdomain can be established. The exclusive subdomain of node i is written
as (D i ), its boundary as (LJ, and the part occupied by both (Di ) and element
(e) is written as (eJ, as shown in Figure 5.10. Equation (6.2.29) will become the
mass balance equation for this subdomain by replacing (D) and (L) with (D i )
and (LJ, respectively. The integral along (Li) can be computed element by
element. Calculating the integrals along the line segments Am and mB in
element (e), we have
= ~ [( Dxx ~~ + Dxy ~~) Substituting the first expressions of (6.2.39a) and (6.2.39b) into this equation,
we obtain:
+ [(Dxxbkj + DxyCkj)(bi - bj) + (Dxybki + DyyCkj)(Ci - c)]Cj
+ [(Dxxbkk + DxyCkk)(bi - bj) + (Dxybkk + DyyCkk)(Ci - c)]Ctl, (6.2.41)
where bl , CI (l = i,j,k) are determined from Eq. (5.1.33). In a similar way, the
second expressions of Eqs. (6.2.39a) and (6.2.39b) are substituted into the
equation to yield:
[ (OC
OC)
(OC
OC)
12 = JinB Dxx ox + Dxy oy dy - DXY oX + D yy oy dx
+ [(Dxx~jj + Dxycjj)(bi - bd + (Dxy~j + DyyCjj)(Ci - Ck)]Cj
+ [(DXX~k + DxAk)(bi - bk) + (Dxybjk + DyAk)(Ci - ck)]Ctl. (6.4.42)
11 plus 12 is just the value of the line integral along (LJ in element (e).
Repeating this process for all elements, which has node i as one of their
vertices, we finally have
= L (IXf;Ci + lXijCj + IXlkCk)'
(6.2.43)
e,