Summary: This paper is concerned with the numerical approximation of hyperbolic systems of balance laws in one space dimension. In order to make easier the obtaining of well-balanced numerical schemes, these systems can be considered as a particular case of hyperbolic nonconservative systems. For these more general problems, the chosen concept of weak solutions is that introduced by G. Dal Maso et al. [J. Math. Pures Appl., IX. Sér. 74, No. 6, 483–548 (1995; Zbl 0853.35068)], which is based on the prescription of a family of paths in the phases space. This prescription is also useful to construct the so-called path-conservative schemes introduced by C. Parés [SIAM J. Numer. Anal. 44, No. 1, 300–321 (2006; Zbl 1130.65089)]. M. J. Castro et al. [J. Comput. Phys. 227, No. 17, 8107–8129 (2008; Zbl 1176.76084)] showed that, for general nonconservative systems, a rather strong convergence assumption is needed to prove a convergence result. Nevertheless, the numerical results obtained there for systems of balance laws suggested that, for that particular case, weaker hypotheses are enough to prove such a result, as we establish here.
In addition, we study the relationship between the well-balanced properties of path-conservative schemes and the prescribed family of paths in this particular case of systems of balance laws.
Finally, we discuss the well-balanced properties of some families of path-conservative schemes for these systems.
For the entire collection see [Zbl 1255.35003].