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This paper addresses the three concepts of \textit{ consistency, stabilityand convergence } in the context of compact finite volume schemes for systemsof nonlinear hyperbolic conservation laws. The treatment utilizes the frameworkof "balance laws". Such laws express the relevant physical conservation laws inthe presence of discontinuities. Finite volume approximations employ thisviewpoint, and the present paper can be regarded as being in this category. Itis first shown that under very mild conditions a weak solution is indeed asolution to the balance law. The schemes considered here allow the computationof several quantities per mesh cell (e.g., slopes) and the notion ofconsistency must be extended to this framework. Then a suitable convergencetheorem is established, generalizing the classical convergence theorem of Laxand Wendroff. Finally, the limit functions are shown to be entropy solutions byusing a notion of "Godunov compatibility", which serves as a substitute to theentropy condition.

Consistency and Convergence of Finite Volume Approximations to Nonlinear Hyperbolic Balance Laws