The first task in solving the deposition problem is to discretize the governing conservation equations. We use finite difference approximations on a non-uniform grid with points numbered by from 1 at the disk boundary to at the inlet boundary.

The "convective" terms, such as in Equation 15–2 are approximated by first-order upwind formulas, as

(15–22)

The velocity is always negative in these problems (flowing from the inlet at large values of toward the substrate at ), so the upwind differencing calls for the derivative to be formed between and . Upwind differences cause substantial artificial diffusion, especially on coarse mesh networks. However, we have found that they lead to much more reliable convergence on coarse meshes early in the iteration procedure. Ultimately, because the meshes are adaptively refined, the artificial diffusion is very small in the final solution.

The first derivative in the summation term in the energy Equation 15–4 is always approximated by a central difference formula,

(15–23)

where the mesh intervals . The coefficients in the summation are evaluated at .

The second-derivative diffusion terms, such as that in the energy equation, are approximated by the following second order central difference:

(15–24)

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

The diffusive terms in the species conservation equations are analogous to the diffusive term in the energy equation. However, because we express them in terms of diffusion velocities, they at first appear different. In Equation 15–5 , the diffusive term is approximated as

(15–25)

The ordinary multicomponent diffusion and thermal diffusion fluxes (Equation 15–10 ) are approximated at the positions as illustrated by

(15–26)

An analogous finite-difference expression is used for the mixture-averaged approximation to the diffusion fluxes. Since the mole fraction of a species can be zero, we avoid difficulties by forming , which is the expression needed in Equation 15–4 , rather than itself ().

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.