3.11 Example
65
sudden change in the gradient of 4 provides a severe test of the discretization
method.
We shall discretize Eq. (3.61) using FD schemes which use the three-point
computational molecule. The resulting algebraic equation at node i reads:
It is common practice to discretize the diffusion term using CDS; thus,
for the outer derivative, we have:
The CDS approximations of the inner derivatives are:
The contributions of the diffusion term to the coefficients of the algebraic
equation (3.64) are thus:
If the convection term is discretized using first-order upwind differences
(UDS - FDS or BDS, depending on the flow direction), we have:
This leads to the following contributions to the coefficients of Eq. (3.64):
Either AC, or A h is zero, depending on the flow direction.
The CDS approximation leads to: