# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [1] Berthon, C.; Coquel, F., Nonlinear projection methods for multi-entropies navier – stokes systems, (), 278-304 · Zbl 1078.76573 [2] Berthon, C.; Coquel, F., Nonlinear projection methods for multi-entropies navier – stokes systems, Math. comput., 76, 1163-1194, (2007) · Zbl 1110.76035 [3] C. Berthon, F. Coquel, P.G. LeFloch, Why many theories of shock waves are necessary. Kinetic relations for nonconservative systems, unpublished notes. · Zbl 1234.35190 [4] F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Frontiers in Math., Birkhäuser Verlag, Bäsel, 2004. · Zbl 1086.65091 [5] Castro, M.J.; Gallardo, J.M.; Parés, C., High-order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. applications to shallow-water systems, Math. comput., 75, 1103-1134, (2006) · Zbl 1096.65082 [6] Castro, M.J.; Gallardo, J.M.; Parés, C., On a well-balanced high-order finite volume scheme for shallow water equations with topography and dry areas, J. comput. phys., 227, 574-601, (2007) · Zbl 1126.76036 [7] Castro, M.J.; García, J.A.; González, J.M.; Macías, J.; Parés, C.; Vázquez, M.E., Numerical simulation of two layer shallow water flows through channels with irregular geometry, J. comput. phys., 195, 202-235, (2004) · Zbl 1087.76077 [8] Castro, M.J.; Macías, J.; Parés, C., A Q-scheme for a class of systems of coupled conservation laws with source term. application to a two-layer 1-D shallow water system, Math. mod. numer. anal., 35, 107-127, (2001) · Zbl 1094.76046 [9] Castro, M.J.; Pardo, A.; Parés, C., Well-balanced numerical schemes based on a generalized hydrostatic reconstruction technique, Math. mod. meth. appl. sci., 17, 2055-2113, (2007) · Zbl 1137.76038 [10] M.J. Castro, A. Pardo, C. Parés, E.F. Toro, in preparation. [11] C. Chalons, F. Coquel, Numerical capture of shock solutions of nonconservative hyperbolic systems via kinetic functions. Analysis and simulation of fluid dynamics, Adv. Math. Fluid Mech., Birkhäuser, Bäsel, 2007, pp. 45-68. · Zbl 1291.35125 [12] Dal Maso, G.; LeFloch, P.G.; Murat, F., Definition and weak stability of nonconservative products, J. math. pures appl., 74, 483-548, (1995) · Zbl 0853.35068 [13] Hayes, B.T.; LeFloch, P.G., Nonclassical shocks and kinetic relations: scalar conservation laws, Arch. rational mech. anal., 139, 1-56, (1997) · Zbl 0902.76053 [14] Hayes, B.T.; LeFloch, P.G., Nonclassical shocks and kinetic relations: finite difference schemes, SIAM J. numer. anal., 35, 2169-2194, (1998) · Zbl 0938.35096 [15] Hou, T.Y.; LeFloch, P.G., Why nonconservative schemes converge to wrong solutions: error analysis, Math. comput., 62, 497-530, (1994) · Zbl 0809.65102 [16] Lax, P.D.; Wendroff, B., Systems of conservation laws, Comm. pure appl. math., 13, 217-237, (1960) · Zbl 0152.44802 [17] LeFloch, P.G., Entropy weak solutions to nonlinear hyperbolic systems in nonconservative form, Comm. partial differen. equat., 13, 669-727, (1988) · Zbl 0683.35049 [18] P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form, Institute for Mathematics and its Application, Minneapolis, Preprint # 593, 1989. [19] P.G. LeFloch, Entropy weak solutions to nonlinear hyperbolic systems in conservative form, in: J. Ballmann, R. Jeltsch (Eds.), Proceedings of the International Conference on Hyperbolic problems, Note on Num. Fluid Mech., vol. 24, Viewieg, Braunschweig, 1989, pp. 362-373. [20] P.G. LeFloch, On some nonlinear hyperbolic problems (in English), Habilitation à Diriger des Recherches, Université de Paris, vol. 6, 1990. [21] P.G. LeFloch, Hyperbolic systems of conservation laws: The theory of classical and nonclassical shock waves, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2002. · Zbl 1019.35001 [22] LeFloch, P.G., Graph solutions of nonlinear hyperbolic systems, J. hyper. differen. equat., 1, 643-689, (2004) · Zbl 1071.35078 [23] LeFloch, P.G.; Liu, T.-P., Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum math., 5, 261-280, (1993) · Zbl 0804.35086 [24] LeFloch, P.G.; Mohamadian, M., Why many shock wave theories are necessary. fourth-order models, kinetic functions, and equivalent equations, J. comput. phys., 227, 4162-4189, (2008) · Zbl 1264.76055 [25] LeFloch, P.G.; Thanh, M.D., The Riemann problem for fluid flows in a nozzle with discontinuous cross-section, Comm. math. sci., 1, 763-796, (2003) · Zbl 1091.35044 [26] LeFloch, P.G.; Thanh, M.D., The Riemann problem for the shallow water equations with discontinuous topography, Comm. math. sci., 5, 865-885, (2007) · Zbl 1145.35082 [27] Muñoz-Ruiz, M.L.; Parés, C., Godunov method for nonconservative hyperbolic systems, Math. method. anal. numer., 41, 169-185, (2007) · Zbl 1124.65077 [28] Noelle, S.; Xing, Y.; Shu, C.W., High-order well-balanced finite volume WENO schemes for shallow water equation with moving water, J. comput. phys., 226, 29-58, (2007) · Zbl 1120.76046 [29] Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. numer. anal., 44, 300-321, (2006) · Zbl 1130.65089 [30] Parés, C.; Castro, M.J., On the well-balance property of roe’s method for nonconservative hyperbolic systems. applications to shallow-water systems, Math. model. numer. anal., 38, 821-852, (2004) · Zbl 1130.76325 [31] J.B. Schijf, J.C. Schonfeld, Theoretical considerations on the motion of salt and fresh water, in: Proceedings of the Minn. Int. Hydraulics Conv., Joint Meeting IAHR Hydro. Div. ASCE. (September 1953) 1953, pp. 321-333. [32] Xing, Y.; Shu, C.-W., High order well-balanced finite volume WENO schemes and discontinuous Galerkin methods for a class of hyperbolic systems with source terms, J. comput. phys., 214, 567-598, (2006) · Zbl 1089.65091
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.