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Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes. (English) Zbl 1176.76084
A number of nonconservative hyperbolic models have been introduced in fluid dynamics to serve as (simplified) models of two-phase or two-layer flows. The authors’ aim, in the present paper, is to address the fundamental question whether finite difference schemes for nonconservative systems converge toward correct weak solutions containing shock waves. Addressing this important issue requires detailed numerical computations, which they carry out here.
For the theory and numerical analysis of nonconservative products, the authors refer to Berthon, Coquel, and LeFloch, who connected the theory of nonconservative products with the concept of a kinetic relation. They introduced a general framework to handle nonconservative systems, which encompasses a large number of examples arising in applications. In particular, they rigorously analyzed a typical model of turbulent fluid dynamics by establishing the existence and properties of a physically relevant family of traveling waves and deriving the corresponding kinetic function.
The use of a numerical strategy based on a direct discretization of the nonconservative system by means of a finite difference scheme, which is formally path-consistent, is advisable and may have the following advantages:
1. The numerical solution is formally consistent with the definition of nonconservative product in the sense of Dal Maso, LeFloch and Murat and, in turn, in the special case when the system admits a conservative subsystem, the numerical scheme is conservative for this subsystem in the sense of Lax.
2. The approximations of the shock provided by the schemes are consistent with a regularization of the system with higher-order terms that vanish as \(\Delta x\) tends to 0. (Obviously, the main drawback is that this regularization depends on the chosen family of paths and on the numerical scheme itself. This issue is dealt with in the present paper.)
3. As originally pointed out by T. Y. Hou, Ph. Rosakis and Ph. LeFloch [ibid. 150, No. 2, 302–331 (1999; Zbl 0936.74052)], in the (simpler) case of scalar hyperbolic equations, the convergence error, measured in terms of convergence error measure or in terms of Hugoniot curves, is noticeable for very fine meshes, for discontinuities of large amplitude, and/or for large-time simulations, only.
4. This strategy is extendable to high-order methods or to multidimensional problems, as developed, together with collaborators, by Coquel and Parés.
The convergence error should also be compared with the experimental error. In the case of the two-layer shallow water system, the shocks captured by Roe scheme and the family of straight lines have been found to be in good agreement with the experimental measurements of internal bores in the Strait of Gibraltar, despite of the simplicity of the family of paths.
In certain special situations, the convergence error measure is found to vanish identically. This is the case of systems whose nonconservative product is associated with a linearly degenerate characteristic field. For more general models, this measure is evaluated very accurately, especially by plotting the shock curves associated with each scheme under consideration; as the authors demonstrate, plotting the shock curves provides a convenient approach for evaluating the validity of a given scheme.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
76L05 Shock waves and blast waves in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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