Construction of Second-Order TVD Schemes for Nonhomogeneous Hyperbolic Conservation Laws

Many of the problems of approximating numerically solutions to nonhomogeneous hyperbolic conservation laws appear to arise from an inability to balance the source and flux terms at steady states. In this paper we present a technique based on the transformation of the nonhomogeneous problem to homogeneous 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 nonhomogeneous conservation laws. However, the application of the numerical methods described for homogeneous conservation laws is not immediate and a new formalization of the classic schemes is required. Particularly, for such cases we extend the explicit, second-order, total variation diminishing schemes of Harten [11]. Numerical test cases in the context of the quasi-one-dimensional flow validate the current schemes, although these schemes are more general and can also be applied to solve other hyperbolic conservation laws with source terms.