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**Numerical analysis of DDFV schemes for semiconductors energy-transport models.**

The energy transport system, [2,3], is composed by two continuity equations (one for the density of electrons, one for the density of internal energy), coupled with a Poisson equation for the electric potential; the two densities depend non-linearly on the unknowns, the chemical potential and the temperature. The key point of the model, presented in

[2], is a change of variables which allows to pass to entropic variables. Thanks to this, it is possible to prove an entropy estimate which gives an a priori estimate on the problem (which leads to the study of regularity and long time behavior of the solution).

We propose, as an extension to some previous works (e.g. [1]), a Discrete Duality Finite Volume scheme (DDFV for short) for the energy transport system; we reproduce at a discrete level the strategy proposed in [2], by proving, as in the continuous case, a discrete entropy-dissipation estimate. We validate our theoretical results with some numerical tests (see [4,5]).

[1] Chainais-Hillairet, C. and Peng, Y.-J., Finite volume scheme for

semiconductor energy-transport model, pp. 139–146, Birkhäuser Basel,

Basel, (2005).

[2] Degond, P. and Génies, S. and Jungel, A., A system of parabolic equa-

tions in nonequilibrium thermodynamics including thermal and electrical

effects, J. Math. Pures Appl. (1997).

[3] Jungel, A., Transport equations for semiconductors, Lecture Notes in

Physics 773 (2009).

[4] Lissoni, G., ”DDFV schemes for semiconductors energy-transport models”,

To appear in Proceeding of the 21th ALGORITMY Conference on Scientific Computing (Podbanske, Slovakia), (2020).

[5] Bessemoulin-Chatard M., Lissoni, G., Mathis H., ”Numerical analysis of DDFV schemes for semiconductors energy-transport models.”, in preparation.