The governing conservation equations require discretization to allow numerical solution. For channel-flow models, a finite difference approximation is used on a non-uniform grid with points numbered as at the lower boundary to at the upper boundary. Approximation of the spatial derivatives is accomplished by finite difference representations on a fixed grid in the normalized stream function.

In the momentum, species, and energy equations, we approximate the second derivatives with conventional central difference formulas as shown in Equation 12–30 .

(12–30)

Here the subscript denotes the j th grid point. We approximate the first derivatives, as needed in Equation 12–3 , by central differences as

(12–31)

We evaluate terms with no derivatives, such as the chemical production rate in Equation 12–2 , using the conditions existing at . Likewise, the coefficients of derivatives, such as in Equation 12–1 , are also evaluated at .

First-order ODE’s, such as Equation 12–16 , are differenced according to the trapezoidal rule as

(12–32)

It is important to represent the integral equations as first-order differential equations and include the variables such as in the dependent variable vector. The reason for this choice is associated with the structure of the Jacobian matrix, which is needed to solve the problem. When Equation 12–32 is used, the number of dependent variables increases, but the Jacobian remains banded (a very desirable feature). On the other hand, if were to be considered as a coefficient in the transport equations, as defined by the integral of the stream function, then the Jacobian loses its banded property and the required computer storage would increase enormously.