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.

With the aid of causal thermodynamics, the Bianchi type-I cosmological model is analyzed under the assumption of plane symmetry. The initial anisotropy gets rapidly extinguished, and for an ample range of values of the parameters describing the cosmic fluid one has inflationary expansion.

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.