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.

The nanometric localization of current transport in heterogeneous Schottky barriers was obtained by the combination of the electric field localization at the apex of a biased conductive atomic force microscopy (c-AFM) tip and of the metal films high-resistivity properties. An abrupt increase of the resistivity, modeled by a quantum-mechanical approach, was measured in Au thin films with a thickness below 10 nm. For Au ultrathin film resistivity, exceeding by two orders of magnitude the bulk value, the nanometric localization of the current transport occurs. This physical effect represents the basic principle of a microscopy approch for two-dimensional Schottky barrier height mapping, which is alternative to conventional ballistic electron emission microscopy (BEEM). A spatial resolution in the order of the tip diameter (10-20 nm) is demonstrated by considering the realistic description of the system (physical and geometrical). Schottky barrier inhomogeneities in a Au/4H-SiC system were imaged with an energy resolution better than 0.1 eV.

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