In the general case, when both diffusive and convective components exist (3.4-32), we use a discretization scheme which is similar to the approach of Scharfetter and Gummel [Sch69] applied in device simulation. Hence, we will sketch its derivation just briefly.
Following Scharfetter and Gummel we integrate the ODE (3.4-35), assuming that the coefficients , and , the electric field , and the flux are constant in the integration interval .
We define the cell Reynolds number (3.4-36) using , the potential difference, and the grid spacing .
In Scharfetter's case the cell Reynolds number was , the ratio of the voltage difference between grid points to the thermal voltage. Solving (3.4-35) and matching to the boundary conditions and , we obtain (3.4-37). is the Bernoulli function (3.4-38).
It is well known that the application of a standard central difference approximation to the flux density would give rise to oscillations if the cell Reynolds number exceeds 2.0 [Bür90].
In the expressions for the fluxes (3.4-33), (3.4-34) we split the velocity into components in - and -direction. Therefore, we use and as projectors, respectively. With this velocity component as value for the cell Reynolds number (3.4-36) and the approximation (3.4-37) we discretize the fluxes (3.4-33) and (3.4-34).