By rewriting the flux equations (2.188) to (2.190)

(3.34) | ||||

(3.35) | ||||

(3.36) |

a common functional form can be recognized. Therefore a general flux equation is introduced

The meanings of the generalized density and temperature is found by inspection:

By projecting eqn. (3.37) onto a grid line a one-dimensional differential equation is obtained

To solve this equation the following assumptions have been made:

- constant general flux ,
- constant electric field

- linear variation of the general temperature

The solution of eqn. (3.41) is found by multiplication with an integrating factor and by sub-sequentially comparing the coefficients of the resulting equation with the total derivative of the product :

(3.44) | ||

(3.45) |

Comparing the coefficients leads to

(3.46) | ||

(3.47) |

This equation can be solved for the integrating factor , taking into account the assumptions (3.42) and (3.43):

(3.48) | ||

(3.49) |

Inserting the integrating factor into eqn. (3.44)

and assuming that the flux is constant between two grid points, eqn. (3.50) can be integrated from to

Commonly eqn. (3.52) is rewritten using the BERNOULLI function

(3.53) |

Beginning with

(3.54) | ||

(3.55) |

and using the abbreviations

(3.56) | ||

(3.57) | ||

(3.58) |

the flux equation can be written as

(3.59) | ||

(3.60) |

or using the BERNOULLI function as

(3.61) | ||

(3.62) |

The concept of assuming a constant flux density was first presented by SCHARFETTER and GUMMEL in the appendix of [51, p.73]. The assumption of a linear variation of the generalized temperature by eqn. (3.43) can be interpreted as a straightforward extension of the SCHARFETTER-GUMMEL scheme.

An advantage of using BERNOULLI functions in the flux equations is that is well defined at .

Inserting the abbreviations (3.38) to (3.40) used for and yields the discretized flux equations

(3.65) | ||

(3.66) |

(3.67) | ||

(3.68) |