Stable upwind schemes for hyperbolic conservation laws with source terms

A class of stable upwind schemes for the numerical solution of nonhomogeneous hyperbolic conservation laws is given. Explicit and implicit-explicit schemes of first and second order accuracy are studies. Stationary solutions are used as data in the first order case. Because of the effect of the source term, the constructed schemes are not TVD (Total Variation Decreasing), but they are locally TVB (Total Variation Bounded). The convergence of the numerical solution towards the entropy solution is proved