Construction of second-order TVD schemes for nonhomogeneous hyperbolic conservation laws

Many of the problems of approximating numerically solutions to nonhomogeneo us hyperbolic conservation laws appear to arise from an inability to balanc e the source and flux terms at steady states. In this paper we present a te chnique based on the transformation of the nonhomogeneous problem to homoge neous form through the definition of a new flux formed by the physical flux and the primitive of the source term. This change preserves the mentioned balance directly and suggests a way to apply well-known schemes to nonhomog eneous conservation laws. However, the application of the numerical methods described for homogeneous conservation laws is not immediate and a new for malization of the classic schemes is required. Particularly, for such cases we extend the explicit, second-order, total variation diminishing schemes of Harten [1 1]. Numerical test cases in the context of the quasi-one-dimen sional flow validate the current schemes, although these schemes are more g eneral and can also be applied to solve other hyperbolic conservation laws with source terms. (C) 2001 Academic Press.