Why many theories of shock waves are necessary. Convergence error in   formally path-consistent schemes
  We are interested in nonlinear hyperbolic systems in nonconservative form arising in fluid dynamics, and, for solutions containing shock waves, we investigate the convergence of finite difference schemes applied to such systems. According to Dal Maso, LeFloch, and Murat's theory, a shock wave theory for a given nonconservative system requires prescribing a priori a family of paths in the phase space. In the present paper, we consider schemes that are formally consistent with a given family of paths, and we investigate their limiting behavior as the mesh is refined. We generalize to systems a property established earlier by Hou and LeFloch for scalar conservation laws, and we prove that nonconservative schemes generate, at the level of the limiting hyperbolic system, a "convergence error" source-term which, provided the total variation of the approximations remains uniformly bounded, is a locally bounded measure. We discuss the role of the equivalent equation associated with a difference scheme; here, the distinction between scalar equations and systems appears most clearly since, for systems, the equivalent equation of a scheme that is formally path-consistent depends upon the prescribed family of paths. The core of this paper is devoted to investigate numerically the approximation of several models arising in fluid dynamics. For systems having nonconservative products associated with linearly degenerate characteristic fields, the convergence error vanishes. For some other models, this measure is evaluated very accurately, especially by plotting the shock curves associated with each scheme under consideration. 
Manuel J. Castro, Philippe G. LeFloch, Mar\'ia Luz Mu\~noz-Ruiz, and   Carlos Par\'es