The first task in solving the flame problem is to discretize the governing conservation equations. We use finite difference approximations on a non-uniform grid with points numbered from 1 at the cold boundary to at the hot boundary. On the convective terms the user has the choice of using either first order windward differences or central differences. Both cases are illustrated using the convective term in the energy equation. The windward difference is given as

(13–26)

where the index refers to the mesh point. The central difference formula is

(13–27)

where . The windward difference formula introduce artificial diffusion on a coarse mesh; this has the effect of spreading out the solution and making the convergence of Newton’s method less sensitive to the starting estimate. However, because the mesh is refined in regions of high gradient, the artificial diffusion becomes relatively unimportant after the solution has progressed to the fine meshes. Nevertheless, for a given mesh, the windward difference approximation is less accurate than the central difference formula. Therefore, the user may want to select the central difference formula on finer meshes or in cases where the solution is converging without difficulty.

The first derivative in the summation term in the energy Equation 13–2 is always approximated by a central difference formula,

(13–28)

and the coefficients in the summation are evaluated at .

The second derivative term in the energy equation is approximated by the following second order central difference:

(13–29)

The coefficients in this formula (at ) are evaluated using the averages of the dependent variables between mesh points.

The diffusive term in the species conservation equation is approximated in a similar way, but it appears to be different because we have written it using diffusion velocities. The ordinary (Equation 13–7 ) and thermal (Equation 13–9 ) diffusion velocities are approximated at the positions as illustrated by the following mixture-averaged evaluation:

(13–30)

and

(13–31)

Since the mole fraction of a species can be zero, we avoid difficulties by forming , which is the expression needed in Equation 13–3 , rather than itself (). After the diffusion velocities are computed at all the mesh midpoints, the correction velocity is computed at the midpoints from

(13–32)

Upon forming the full diffusion velocities the diffusion term is evaluated with the following difference approximation.

(13–33)

All the non-differentiated terms, such as the chemical production rate terms, are evaluated at the mesh points . Coefficients not appearing within derivatives are also evaluated at the mesh points.