Central Schemes for Systems of Balance Laws

Several models in mathematical physics are described by quasilin ear hyperbolic systems with source term which in several cases may become sti Here a suitable central numerical scheme for such problems is developed and application to shallow water equations Broadwell model and Extended Thermodynamics are mentioned The numerical methods are a generalization of the Nessyahu Tadmor scheme to the non homogeneous case We propose two ways for treating the production term The rst is obtained by including the cell averages of the productions while the second family of schemes is obtained by a splitting strategy Introduction In several problems of mathematical physics hyperbolic systems of balance laws arise In particular we mention hyperbolic systems with relaxation such as discrete velocity models in kinetic theory gas with vibrational degrees of freedom hydrodynamical models for semiconductors radiation hydrodynamics Lately the development of high order shock capturing methods for conserva tion laws has become an interesting area or research However most schemes deal almost exclusively with the homogeneous case The extension to systems with a source term has been studied in where a method of line approach to gether with splitting techniques has been used Here we consider the extension of second order central schemes to the non homogeneous case The aim is to provide a general purpose robust scheme for systems of balance laws The main advantage of central schemes is their exibility in fact they do not require the knowledge of the characteristic structure of the system and the exact or approximate solution to the Riemann problem at variance with upwind based schemes There are systems with relaxation for which the analytical expression of the eigenvalues and eigenvectors is not known Typical examples are given by monoatomic gas in Extended Thermodynamics and some hydrodynamical models for electron transport in semiconductors Explicit central schemes for balance laws with source have been considered in and applied to the shallow water equations