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5. Finite Element Methods for Solving Hydrodynamic Dispersion Equations
integral on the left-hand side of Eq. (5.2.3), we then obtain
= I (<'X~Ci + <'XijCj + <'XfkCk),
(5.2.9)
ei
where
me .
<' Xii = - 4d' [D~xbib/ + D~y(Cib/ + biC/) + D;yCic/], (I = i,j, k) (5.2.10)
e
1
mei = 24 (lOmi + 7mj + 7md·
(5.2.11)
Lei is the sum over an the elements with node i as one of their vertices, and
D~x, D~y, D;y are values of dispersion coefficients of element (e).
For the second line integral on the left-hand side of Eq. (5.2.3), we have
r mC(Yydx- Y"dy) = -fr [: (VxmC) + : (YymC)] dx dy
J(L)
J(D) uX
uy
= - f i D) (mvx ~~ + mYy ~~)dXdY
-f iD) c[O(;;J + 0(;;)] dx dy. (5.2.12)
Calculating the first integral on the right-hand side, we get
where
1
(Ji~ = 432 [(170mi + 47mj + 47mk)Vxi
+ (47mi + 23mj + 14mk)Vxj + (47mi + 14mj + 23mk)Vxk ],
(J.e = _1_[(170m. + 47m· + 47m k )V.
'y
432
'
J
y'
and where V xi , Yyi, Vxj , Yyj, Vxk , Yyk are the values of velocity components Vx,
Yy at nodes i, j, k, respectively.