Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T09:30:53.400Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Compressible fluids and gas dynamics, general Hyperbolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  08 March 2017

Abdelaziz Beljadid ,
Philippe G. LeFloch ,
Siddhartha Mishra  and
Carlos Parés
Abdelaziz Beljadid*
Affiliation:
Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, USA
Philippe G. LeFloch*
Affiliation:
Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Université Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris, France
Siddhartha Mishra*
Affiliation:
Seminar for Applied Mathematics (SAM), ETH Zurich, Rämistrasse-101, Zürich, 8092, Switzerland
Carlos Parés*
Affiliation:
Departamento de Análisis Matemático, Universidad de Málaga, 29071 Málaga, Spain
*
*Corresponding author. Email addresses:[email protected] (A. Beljadid), [email protected] (P. G. LeFloch), [email protected] (S. Mishra), [email protected] (C. Parés)
*Corresponding author. Email addresses:[email protected] (A. Beljadid), [email protected] (P. G. LeFloch), [email protected] (S. Mishra), [email protected] (C. Parés)
*Corresponding author. Email addresses:[email protected] (A. Beljadid), [email protected] (P. G. LeFloch), [email protected] (S. Mishra), [email protected] (C. Parés)
*Corresponding author. Email addresses:[email protected] (A. Beljadid), [email protected] (P. G. LeFloch), [email protected] (S. Mishra), [email protected] (C. Parés)
Get access

Abstract

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.

Keywords

Nonlinear hyperbolicnonconservative systemdiscrete schemecontrolled dissipationcomplex fluid dynamics

MSC classification

Secondary: 35L65: Conservation laws 65M06: Finite difference methods 76N15: Gas dynamics, general
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 4 , April 2017 , pp. 913 - 946
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Abgrall, R. and Karni, S., A comment on the computation of nonconservative products, J. Comp. Phys. 229 (2010), 2759–2763.Google Scholar
[2] Allouges, F. and Merlet, B.. Approximate shock curves for non-conservative hyperbolic systems in one space dimension. J. Hyp. Diff. Eq. 1 (2004), 769–788.Google Scholar
[3] Berthon, C., Nonlinear scheme for approximating a nonconservative hyperbolic system, C. R. Math. Acad. Sci. Paris 335 (2002), 1069–1072.Google Scholar
[4] Berthon, C., Boutin, B., and Turpault, R., Shock profiles for the shallow-water Exner models, Adv. Applied Math. Mech. (2016), to appear.Google Scholar
[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. 307–314.Google Scholar
[6] Berthon, C. and Coquel, F., Nonlinear projection methods for multi-entropies Navier–Stokes systems, Math. Comp. 76 (2007), 1163–1194.Google Scholar
[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), 1–37.CrossRefGoogle Scholar
[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), 8107–8129.Google Scholar
[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), 1371–1391.Google Scholar
[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), 1103–1134.Google Scholar
[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), 107–127.Google Scholar
[12] Castro, M.J., Parés, C., Puppo, G., and Russo, G., Central schemes for nonconservative hyperbolic systems, SIAM J. Sci. Comput. 34 (2012), 523–558.Google Scholar
[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. 45–68.Google Scholar
[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), 1–23.Google Scholar
[15] Dal Maso, G., LeFloch, P.G., and Murat, F., Definition and weak stability of nonconservative products, J. Math. Pures Appl. 74 (1995), 483–548.Google Scholar
[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), 1731–1748.Google Scholar
[17] Ernest, J., LeFloch, P.G., and Mishra, S., Schemes with well-controlled dissipation, SIAM J. Numer. Anal. 53 (2015), 674–699.CrossRefGoogle Scholar
[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), 117–140.Google Scholar
[19] Fjordholm, U. S. and Mishra, S., Accurate numerical discretizations of nonconservative hyperbolic systems, M2AN: Math. Model. Numer. Anal. 46 (2012), 187–296.Google Scholar
[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), 339–365.Google Scholar
[21] Hayes, B.T. and LeFloch, P.G., Nonclassical shocks and kinetic relations: Finite difference schemes, SIAM J. Numer. Anal. 35 (1998), 2169–2194.Google Scholar
[22] Hou, T.Y. and LeFloch, P.G., Why nonconservative schemes converge to wrong solutions: Error analysis, Math. Comp. 62 (1994), 497–530.Google Scholar
[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), 302–331.Google Scholar
[24] Karni, S.. Viscous shock profiles and primitive formulations, SIAM J. Num. Anal. 29 (1992), 1592–1609.Google Scholar
[25] LeFloch, P.G., Entropy weak solutions to nonlinear hyperbolic systems in nonconservative form, Comm. Part. Diff. Equ. 13 (1988), 669–727.Google Scholar
[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: https://conservancy.umn.edu/bitstream/handle/11299/5107/593.pdf Google Scholar
[27] LeFloch, P.G., On some nonlinear hyperbolic problems, Memoir of “Habilitation à Diriger des Recherches”, Université Pierre et Marie Curie, Paris, July 1990.Google Scholar
[28] LeFloch, P.G., Propagating phase boundaries: Formulation of the problemand existence via the Glimm scheme, Arch. Ration. Mech. Anal. 123 (1993), 153–197.CrossRefGoogle Scholar
[29] LeFloch, P.G., Hyperbolic Systems of Conservation Laws. The theory of classical and nonclassical shock waves, Lectures in Mathematics, ETH Zürich, Birkhäuser, 2002.Google Scholar
[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) 237–272.Google Scholar
[31] LeFloch, P.G. and Liu, T.-P., Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum Math. 5 (1993), 261–280.Google Scholar
[32] LeFloch, P.G. and Mishra, S.. Numerical methods with controlled dissipation for small-scale dependent shocks. Acta Num. 23 (2014), 743–816.Google Scholar
[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), 4162–4189.Google Scholar
[34] LeFloch, P.G. and Rohde, C., High-order schemes, entropy inequalities, and nonclassical shocks, SIAM J. Numer. Anal. 37 (2000), 2023–2060.Google Scholar
[35] Parés, C., Numerical methods for nonconservative hyperbolic systems: a theoretical framework, SIAM J. Num. Anal. 44 (2006), 300–321.Google Scholar
[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. 67–122.Google Scholar
[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), 23–52.Google Scholar
[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), 297–333.Google Scholar
[39] Shu, C.-W. and Osher, S.. Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comp. Phys. 77 (1988), 439–471.Google Scholar
[40] Stewart, H. B. and Wendroff, B., Two-phase flow: models and methods, J. Comput. Phys. 56 (1984), 363–409.Google Scholar
[41] Whitham, G.B.. Linear and Nonlinear Waves. John Wiley and Sons, 2011.Google Scholar