We call a flux purely convective if the diffusive part, driven by the concentration gradient, vanishes, i.e. . This lack of dissipation may cause serious numerical troubles. The resulting PDE equation is not a second order differential equation.
Here denotes a velocity. A central difference approximation of this term in the form , using linearly interpolated values and , is known to produce oscillations or wiggles in the solution [Mit80] - convergence cannot be guaranteed.
Remedy is expected from upwinding schemes, see for instance [Kre87], where the -term is approximated in the form . In the limit the upwinding parameter is for and for .
For the partial derivatives in (3.4-29), (3.4-30) we apply again the difference approximations (3.4-24) and (3.4-25). The values of the electrostatic potential and the metric coefficients , , , and are available at the grid points. The mobility and the velocity (from other driving forces) that are explicitly available at the center point and may depend on all concentrations.