68
3. Finite Difference Methods
Exact
.8
------------- Calculated
Fig. 3.10. Solution of the 1D convection/diffusion equation at Pe = 50 using CDS
(left) and UDS (right) for convection terms and a non-uniform grid with 11 nodes
(grid dense near right end)
an expansion factor re = 0.7, see Eq. (3.21). The minimum Peclet number is
thus Pemi, = 0.625 near right boundary, and the maximum is Pe,,,
= 15.5
near left boundary. The Peclet number is thus smaller than two in the region
where 4 undergoes strong change and is large in the region of nearly constant
4. The calculated profiles on this grid using the CDS and UDS schemes are
shown in Fig. 3.10. There are no oscillations in the CDS solution. Moreover,
it is as accurate as the solution on a uniform grid with four times as many
nodes. The accuracy of the UDS solution has also been improved by using a
non-uniform grid but is still unacceptable.
- CDS, Nonuniform
UDS, Nonuniform
I
Fig. 3.11. Average error in the solution
of the 1D convection/diffusion equation
for Pe=50 as a function of the average
mesh spacing