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Schemes with Well-Controlled Dissipation. Hyperbolic Systems in Nonconservative Form

  • Abdelaziz Beljadid (a1), Philippe G. LeFloch (a2), Siddhartha Mishra (a3) and Carlos Parés (a4)

We propose here a class of numerical schemes for the approximation of weak solutions to nonlinear hyperbolic systems in nonconservative form—the notion of solution being understood in the sense of Dal Maso, LeFloch, and Murat (DLM). The proposed numerical method falls within LeFloch-Mishra's framework of schemes with well-controlled dissipation (WCD), recently introduced for dealing with small-scale dependent shocks. We design WCD schemes which are consistent with a given nonconservative system at arbitrarily high-order and then analyze their linear stability. We then investigate several nonconservative hyperbolic models arising in complex fluid dynamics, and we numerically demonstrate the convergence of our schemes toward physically meaningful weak solutions.

Corresponding author
*Corresponding author. Email addresses: (A. Beljadid), (P. G. LeFloch), (S. Mishra), (C. Parés)
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[1] Abgrall R. and Karni S., A comment on the computation of nonconservative products, J. Comp. Phys. 229 (2010), 27592763.
[2] Allouges F. and Merlet B.. Approximate shock curves for non-conservative hyperbolic systems in one space dimension. J. Hyp. Diff. Eq. 1 (2004), 769788.
[3] Berthon C., Nonlinear scheme for approximating a nonconservative hyperbolic system, C. R. Math. Acad. Sci. Paris 335 (2002), 10691072.
[4] Berthon C., Boutin B., and Turpault R., Shock profiles for the shallow-water Exner models, Adv. Applied Math. Mech. (2016), to appear.
[5] Berthon C. and Coquel F., Nonlinear projection methods for multi-entropies Navier–Stokes systems. In “Finite Volumes for Complex Applications II: Problems and Perspectives”, Hermes Science Publ., 1999, pp. 307314.
[6] Berthon C. and Coquel F., Nonlinear projection methods for multi-entropies Navier–Stokes systems, Math. Comp. 76 (2007), 11631194.
[7] Berthon C., Coquel F., and LeFloch P.G., Why many theories of shock waves are necessary: Kinetic relations for nonconservative systems, Proc. Royal Soc. Edinburgh 137 (2012), 137.
[8] Castro M. J., LeFloch P.G., Muñoz-Ruiz M.L., and Parés C., Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes, J. Comput. Phys. 227 (2008), 81078129.
[9] Castro M.J., Fjordholm U.S., Mishra S., and Parés C., Entropy conservative and entropy stable schemes for nonconservative hyperbolic systems, SIAM J. Numer. Anal. 51 (2013), 13711391.
[10] Castro M.J., Gallardo J.M., and 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. Comp. 75 (2006), 11031134.
[11] Castro M.J, Macías J., and 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, M2AN: Math. Model. Numer. Anal. 35 (2001), 107127.
[12] Castro M.J., Parés C., Puppo G., and Russo G., Central schemes for nonconservative hyperbolic systems, SIAM J. Sci. Comput. 34 (2012), 523558.
[13] Chalons C. and Coquel F., Numerical capture of shock solutions of nonconservative hyperbolic systems via kinetic functions. In Analysis and Simulation of Fluid Dynamics, Advances in Mathematical Fluid Mechanics, Birkhäuser, 2007, pp. 4568.
[14] Chalons C. and LeFloch P.G., High-order entropy conservative schemes and kinetic relations for van der Waals fluids, J. Comput. Phys. 167 (2001), 123.
[15] Dal Maso G., LeFloch P.G., and Murat F., Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995), 483548.
[16] Dumbser M., Castro M.J., Parés C., and Toro E.F., ADER schemes on unstructured meshes for nonconservative hyperbolic systems: Applications to geophysical flows, Comp. & Fluids 38 (2009), 17311748.
[17] Ernest J., LeFloch P.G., and Mishra S., Schemes with well-controlled dissipation, SIAM J. Numer. Anal. 53 (2015), 674699.
[18] Fernández-Nieto E.D., Castro M.J., and Parés C.. On an intermediate field capturing Riemann solver based on a parabolic viscosity matrix for the two-layer shallow water system. J. Sci. Comp. 48 (2011), 117140.
[19] Fjordholm U. S. and Mishra S., Accurate numerical discretizations of nonconservative hyperbolic systems, M2AN: Math. Model. Numer. Anal. 46 (2012), 187296.
[20] Gosse L., A well-balanced scheme using nonconservative products designed for hyperbolic systems of conservation laws with source terms, Math. Mod. Meth. Appl. Sci. 11 (2001), 339365.
[21] Hayes B.T. and LeFloch P.G., Nonclassical shocks and kinetic relations: Finite difference schemes, SIAM J. Numer. Anal. 35 (1998), 21692194.
[22] Hou T.Y. and LeFloch P.G., Why nonconservative schemes converge to wrong solutions: Error analysis, Math. Comp. 62 (1994), 497530.
[23] Hou T.Y., Rosakis P., and LeFloch P.G., A level set approach to the computation of twinning and phase transition dynamics, J. Comput. Phys. 150 (1999), 302331.
[24] Karni S.. Viscous shock profiles and primitive formulations, SIAM J. Num. Anal. 29 (1992), 15921609.
[25] LeFloch P.G., Entropy weak solutions to nonlinear hyperbolic systems in nonconservative form, Comm. Part. Diff. Equ. 13 (1988), 669727.
[26] LeFloch P.G., Shock waves for nonlinear hyperbolic systems in nonconservative form, Institute for Math. and its Appl., Minneapolis, Preprint # 593, 1989. Available at:
[27] LeFloch P.G., On some nonlinear hyperbolic problems, Memoir of “Habilitation à Diriger des Recherches”, Université Pierre et Marie Curie, Paris, July 1990.
[28] LeFloch P.G., Propagating phase boundaries: Formulation of the problemand existence via the Glimm scheme, Arch. Ration. Mech. Anal. 123 (1993), 153197.
[29] LeFloch P.G., Hyperbolic Systems of Conservation Laws. The theory of classical and nonclassical shock waves, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2002.
[30] LeFloch P.G., Kinetic relations for undercompressive shock waves: Physical, mathematical, and numerical issues. In Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena, Vol. 526 of Contemporary Mathematics, AMS (2010) 237272.
[31] LeFloch P.G. and Liu T.-P., Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Math. 5 (1993), 261280.
[32] LeFloch P.G. and Mishra S.. Numerical methods with controlled dissipation for small-scale dependent shocks. Acta Num. 23 (2014), 743816.
[33] LeFloch P.G. and Mohamadian M., Why many shock wave theories are necessary. Fourth-order models, kinetic functions, and equivalent equations, J. Comput. Phys. 227 (2008), 41624189.
[34] LeFloch P.G. and Rohde C., High-order schemes, entropy inequalities, and nonclassical shocks, SIAM J. Numer. Anal. 37 (2000), 20232060.
[35] Parés C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. Num. Anal. 44 (2006), 300321.
[36] Parés C., Path-conservative numerical methods for nonconservative hyperbolic systems, In Numerical methods for balance laws, Vol. 24, Quaderni di matematica, Dipto. di Matematica della Seconda Universitá di Napoli, 2009, pp. 67122.
[37] Parés C. and Muñoz M.L.. On some difficulties of the numerical approximation of nonconservative hyperbolic systems, Bol. Soc. Esp. Mat. Apl. 47 (2009), 2352.
[38] Sainsaulieu L. and Raviart P.-A., A nonconservative hyperbolic system modeling spray dynamics: solution of the Riemann problem, Math. Models Methods Appl. Sci. 5 (1995), 297333.
[39] Shu C.-W. and Osher S.. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comp. Phys. 77 (1988), 439471.
[40] Stewart H. B. and Wendroff B., Two-phase flow: models and methods, J. Comput. Phys. 56 (1984), 363409.
[41] Whitham G.B.. Linear and Nonlinear Waves. John Wiley and Sons, 2011.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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