182
6. Numerical Solutions of Advection-Dominated Problems
3. Project the concentrations of the moving points onto the fixed nodt:s. This
can be done by using the simple arithmetic me an, distance weighted mean
or an interpolation method. The concentrations of the nodes thus obtained are written as {ct1 1 };
_ _
4. Define ct1 1 to be the weighted mean between ct? and ct?, i.e., let
(6.4.13)
where ° < A < 1. The weighting parameter A is taken to be close to zero in
elements occupied by a steep front because these elements contain more
moving points and ct? is rather accurate. For elements located in a
smooth front, A is chosen to be close to one because there are fewer
moving points and CU 1 is more reliable;
5. Increase or reduce the number of moving points in each element according to the criteria given above.
6.4.3 The Hybrid Moving Point-Characteristics Finite
Element M ethod
For the standard Galerkin finite element procedure, we assume the approximate solution
N
C(x, y, t) ;:;::: L Cß)(h(x, y),
j;1
and require that it satisfy
f r [DC - V' (DVC)] rPidR = 0,
J(R)
Dt
(i = 1,2, ... ,N).
(6.4.14)
(6.4.15)
Eliminating the se co nd order derivative term by using Green's formula, we
have
f r DC rPidR + fr DVC'VrPidR - r g2rPidr = 0,
J(R) Dt
J(R)
J(r>
(i = 1,2, .. . ,N).
(6.4.16)
In the above equation, boundary condition, - DV C . n = g 2' has been used
along boundary (r).
The basis function rPi(X, y) are non-zero only in the elements in volving
node i. For instance, if triangle elements and linear basis functions are
adopted, rPi(X, y) are non-zero only in those tri angle elements which contain
node i as their common vertex. Node i is located almost at the center of the
subdomain. So approximately, we have
ff
DC
DCiff
-rPidR=rPi dR,
(R) Dt
Dt
(R)
(6.4.17)