Following questions raised at a (very interesting) conference I attended at Bilbao in September 2009 on hyperbolic systems with source-terms and non-conservative form, I want to emphasize here a fact of importance in numerical computations for such problems.

Consider the issue of computing shock waves to nonconservative hyperbolic systems, as well as to hyperbolic conservation laws generated by zero diffusive-dispersive limits,  In these two instance, an additional selection criterion beyond an entropy inequality is necessary, that is, a kinetic relation for the latter and a family of paths (or one or several kinetic functions) for the former. The shocks under consideration turn out to strongly depend on small scales that are represented by higher-order terms (viscosity, heat conduction, capillarity, etc) that must be included in an associated augmented model. No universal selection criterion should be sought for, but rather a different theory of shock waves is necessary in each physical context.

The key point to stress is that no finite difference schemes (say expressed as smooth algebraic expressions of the coefficients of the conservative or nonconservative equations) has never been found to converge to correct weak solutions ! This difficulty was observed in Hou and LeFloch in 1993 for nonconservative systems and in Hayes and LeFloch in 1994 for diffusive-dispersive limits.

The point is that in both the continuous model and the discrete scheme, small scale features are critical to the selection of shocks. The balance between diusive and dispersive features determines which shocks are selected. In nonconservative systems, the competition takes place between the (hyperbolic) propagation part and the (viscous) regularization. These small scale features cannot be quite the same at the continuous and at the discrete levels, since a continuous dynamical system of ordinary dierential equations cannot be exactly represented by a discrete dynamical system of finite dierence equations. Strictly speaking, finite difference schemes do not converge to the correct weak solution when small-scaled are the driving factor for the selection of shock waves.

Still, the good news is that schemes can be developed that approach the exact solutions as much as one requests. The design of suitable schemes is achieved via an analysis of the equivalent equation associated with a scheme by formal Taylor expanding its coefficients. The accuracy of the numerical solution improves as the equivalent equation coincides with the augmented model at a higher and higher order of approximation. For recent developments, we refer the interested reader to the following two papers.

P.G. LeFloch and M. Mohamadian, Why many shock wave theories are necessary. Fourth-order models, kinetic functions, and equivalent equations, J. Comput. Phys. 227 (2008), 4162–4189.

Abstract. We consider several systems of nonlinear hyperbolic conservation laws describing the dynamics of nonlinear waves in presence of phase transition phenomena. These models admit under-compressive shock waves which are not uniquely determined by a standard entropy criterion but must be characterized by a kinetic relation. Building on earlier work by LeFloch and collaborators, we investigate the numerical approximation of these models by high-order finite difference schemes, and uncover several new features of the kinetic function associated with physically motivated second and third-order regularization terms, especially viscosity and capillarity terms. On one hand, the role of the equivalent equation associated with a finite difference scheme is discussed. We conjecture here and demonstrate numerically that the (numerical) kinetic function associated with a scheme approaches the (analytic) kinetic function associated with the given model – especially since its equivalent equation approaches the regularized model at a higher order. On the other hand, we demonstrate numerically that a kinetic function can be associated with the thin liquid film model and the generalized Camassa–Holm model. Finally, we investigate to what extent a kinetic function can be associated with the equations of van der Waals fluids, whose flux-function admits two inflection points.

M.J. Castro, P.G. LeFloch, M.L. Munoz-Ruiz, and C. Pares, Why many theories of shock waves are necessary. Convergence error in formally path-consistent schemes, J. Comput. Phys. 227 (2008), 8107–8129.

Abstract. 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 first 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, an {\sl convergence error source-term} which, provided the total variation of the approximations remains uniformly bounded, is a locally bounded measure. This convergence error measure is supported on the shock trajectories and, as we demonstrate here, is usually “small”. In the special case that the scheme converges in the sense of graphs —a rather strong convergence property often violated in practice— then this measure source-term vanishes. We also 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 (simplified or full) hyperbolic models arising in fluid dynamics. This leads us to the conclusion that for systems having nonconservative products associated with linearly degenerate characteristic fields, the convergence error vanishes. For more general models, this measure is evaluated very accurately, especially by plotting the shock curves associated with each scheme under consideration; as we demonstrate, plotting the shock curves provide a convenient approach for evaluating the range of validity of a given scheme.