4. 1. 8 Refinement of Models

A method which is typically used in scientific computing is the refinement of formulae which have been proven to work. As an example one begins with a linear differential equation for instance the Poisson equation. Then one adds the isothermal drift diffusion current relations and performs the necessary testing. Afterwards, the temperature is added using another solution quantity and the non-isothermal drift-diffusion model is implemented.

In all stages of the implementation additional derivatives of residua with respect to other values have to be considered. If many different solution variables are involved, this requires an effort which approximately depends quadratically on the number of the solution variables. Even if done aided by computational algebra tools, this is very cumbersome and often results in oversights and flaws which are difficult to find.

The use of linearized equations offers the advantage that all derivatives are implicitly calculated and adding further solution functions does not imply further efforts, except the formulation of the residuum of the new governing equation. All derivatives with respect to the other variables are implicitly determined.