Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes

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 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.

A depth-averaged debris-flow model that includes the effects of evolving dilatancy. II. Numerical predictions and experimental tests

“…Non-conservative systems, in which W x,y (q) = 0 applies, can pose additional challenges related to determining generalized discontinuous solutions, which cannot be based on standard Rankine-Hugoniot jump conditions derived from conservation laws (e.g. [18,19] and appendix B). Desirable stability properties and well-posedness of systems having the form of (2.19) (neglecting the source term ϕ) are established by hyperbolicity in the eigenstructure of A x (q) and A y (q) (e.g.…”

On some fast well-balanced first order solvers for nonconservative systems

“…The Lax-Friedrichs method written in this form has been introduced and analyzed in [6]. From equation (3.13), it is clear that the method is independent of the particular choice of Roe linearization.…”

“…In [6] it has been proved that, in general, the numerical solutions provided by a path-conservative numerical scheme converge to functions which solve a perturbed system in which an error source-term appears on the right-hand side. The appearance of this source-term, which is a measure supported on the discontinuities, has been first observed in [15] when a scalar conservation law is discretized by means of a nonconservative numerical method.…”

“…Nevertheless, in certain special situations the convergence error vanishes for finite difference methods: this is the case for systems of balance laws. Moreover for more general problems, even when the convergence error is present, it may only be noticeable for very fine meshes, for discontinuities of large amplitude, and/or for large-time simulations; see [6] for details.…”

“…In the first paragraph, we introduce a family of numerical methods that generalizes the conservative Rusanov schemes. The Lax-Friedrichs method for nonconservative systems introduced and analyzed in [6] can be considered as a particular case. As their conservative counterpart, these methods satisfy a discrete entropy inequality and they are positivity-preserving.…”