Several models in mathematical physics are described by quasi-linear hyperbolic systems with source term and in several cases the production term can become stiff. Here suitable central numerical schemes for such problems are developed and applications to the Broadwell model and extended thermodynamics are presented. The numerical methods are a generalization of the Nessyahu-Tadmor scheme to the nonhomogeneous case by including the cell averages of the production terms in the discrete balance equations. A second order scheme uniformly accurate in the relaxation parameter is derived and its properties analyzed. Numerical tests confirm the accuracy and robustness of the scheme.

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A hydrodynamical model based on the theory of extended thermodynamics is presented for carrier transport in semiconductors. Closure relations for fluxes are obtained by employing the maximum entropy principle. The production terms are modeled by fitting the Monte Carlo data for homogeneously doped semiconductors. The mathematical properties of the model are studied. A suitable numerical method, which is a generalization of the Nessyahu-Tadmor scheme to the nonhomogeneous case, is provided. The validity of the constitutive relations has been assessed by comparing the numerical results with detailed Monte Carlo simulations.

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SUMMARYA consistent hydrodynamical model for electron transport in silicon semiconductors, free of any ÿt-ting parameter, has been formulated in Anile and Romano (Continuum Mechanics Thermodynamics 1999; 11:307-325) and Romano (Continuum Mechanics Thermodynamics 1999; 12:31-51) on the basis of the maximum entropy principle, by considering the energy band described by the Kane dispersion relation. Explicit constitutive functions for uxes and production terms in the macroscopic balance equations of density, crystal momentum, energy and energy ux have been obtained. Scatterings of electrons with non-polar optical phonons (both for intervalley and intravalley interactions), acoustic phonons and impurities have been taken into account.In this article we show the link with other macroscopic models describing the motion of charge carriers. In particular, under suitable scaling assumptions, an energy transport model is recovered. An analysis of the formal properties is given by showing that the evolution equations form a hyperbolic system in the physically relevant region of the space of the dependent variables. At last, by using the numerical method developed in Liotta et al.

Both the truncated and the full version of the causal thermodynamic theory of nonequilibrium phenomena are used to study the evolution of a Bianchi type-I11 cosmological model. The initial anisotropy dies away rapidly and the de Sitter solutions are stable while the Friedmann ones are not.PACS number(s): 98.80.Hw, 04.40.Nr

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