Hostname: page-component-7c8c6479df-ph5wq Total loading time: 0 Render date: 2024-03-28T12:47:12.035Z Has data issue: false hasContentIssue false

Numerical methods with controlled dissipation for small-scale dependent shocks*

Published online by Cambridge University Press:  12 May 2014

Philippe G. LeFloch
Affiliation:
Laboratoire Jacques-Louis Lions, Centre National de la Recherche Scientifique, Université Pierre et Marie Curie, 4 Place Jussieu, 75258 Paris, France, E-mail: contact@philippelefloch.org
Siddhartha Mishra
Affiliation:
Seminar for Applied Mathematics, D-Math, Eidgenössische Technische Hochschule, HG Raemistrasse, Zürich–8092, Switzerland, E-mail: siddhartha.mishra@sam.math.ethz.ch

Extract

We provide a ‘user guide’ to the literature of the past twenty years concerning the modelling and approximation of discontinuous solutions to nonlinear hyperbolic systems that admit small-scale dependent shock waves. We cover several classes of problems and solutions: nonclassical undercompressive shocks, hyperbolic systems in nonconservative form, and boundary layer problems. We review the relevant models arising in continuum physics and describe the numerical methods that have been proposed to capture small-scale dependent solutions. In agreement with general well-posedness theory, small-scale dependent solutions are characterized by a kinetic relation, a family of paths, or an admissible boundary set. We provide a review of numerical methods (front-tracking schemes, finite difference schemes, finite volume schemes), which, at the discrete level, reproduce the effect of the physically meaningful dissipation mechanisms of interest in the applications. An essential role is played by the equivalent equation associated with discrete schemes, which is found to be relevant even for solutions containing shock waves.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2014 

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.)

Footnotes

*

Colour online for monochrome figures available at journals.cambridge.org/anu.

References

REFERENCES

Abeyaratne, R. and Knowles, J. K. (1991a), ‘Kinetic relations and the propagation of phase boundaries in solids’, Arch. Ration. Mech. Anal. 114, 119154.CrossRefGoogle Scholar
Abeyaratne, R. and Knowles, J. K. (1991b), ‘Implications of viscosity and strain-gradient effects for the kinetics of propagating phase boundaries in solids’, SIAM J. Appl. Math. 51, 12051221.CrossRefGoogle Scholar
Abgrall, R. and Karni, S. (2010), ‘A comment on the computation of nonconservative products’, J. Comput. Phys. 45, 382403.Google Scholar
Abgrall, R. and Karni, S. (2013), ‘High-order unstructured Lagrangian one-step WENO finite volume schemes for nonconservative hyperbolic systems: Applications to compressible multi-phase flows’, Comput. Fluids 86, 405432.Google Scholar
Adimurthi, , Mishra, S. and Gowda, G. D. (2005), ‘Optimal entropy solutions for conservation laws with discontinuous flux-functions’, J. Hyperbolic Differ. Equ. 2, 783837.CrossRefGoogle Scholar
Amadori, D. (1997), ‘Initial-boundary value problems for nonlinear systems of conservation laws’, NoDEA: Nonlinear Differ. Equ. Appl. 4, 142.Google Scholar
Amadori, D. and Colombo, R. M. (1997), ‘Continuous dependence for 2 × 2 conservation laws with boundary’, J. Differ. Equ. 138, 229266.Google Scholar
Amadori, D., Baiti, P., LeFloch, P. G. and Piccoli, B. (1999), ‘Nonclassical shocks and the Cauchy problem for nonconvex conservation laws’, J. Differ. Equ. 151, 345372.Google Scholar
Ancona, F. and Bianchini, S. (2006), Vanishing viscosity solutions of hyperbolic systems of conservation laws with boundary. In WASCOM 2005: 13th Conference on Waves and Stability in Continuous Media (Monaco, R.et al., eds), World Scientific, pp. 1321.Google Scholar
Ancona, F. and Marson, A. (1999), ‘Scalar non-linear conservation laws with integrable boundary data’, Nonlinear Anal. 35, 687710.Google Scholar
Antonelli, P. and Marcati, P. (2009), ‘On the finite energy weak solutions to a system in quantum fluid dynamics’, Comm. Math. Phys. 287, 657686.Google Scholar
Arvanitis, C., Makridakis, C. and Sfakianakis, N. I. (2010), ‘Entropy conservative schemes and adaptive mesh selection for hyperbolic conservation laws’, J. Hyperbolic Differ. Equ. 7, 383404.Google Scholar
Audebert, B. and Coquel, F. (2006), Hybrid Godunov–Glimm method for a nonconservative hyperbolic system with kinetic relations. In Numerical Mathematics and Advanced Applications: Proc. ENUMATH 2005, Springer, pp. 646653.CrossRefGoogle Scholar
Bachmann, F. and Vovelle, J. (2006), ‘Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients’, Comm. Partial Differ. Equ. 31, 371395.Google Scholar
Baiti, P., LeFloch, P. G. and Piccoli, B. (1999), Nonclassical shocks and the Cauchy problem: General conservation laws. In Nonlinear Partial Differential Equations, Vol. 238 of Contemporary Mathematics, AMS, pp. 125.Google Scholar
Baiti, P., LeFloch, P. G. and Piccoli, B. (2000), BV stability via generalized characteristics for nonclassical solutions of conservation laws. In EQUADIFF 99: Proc. International Conference on Differential Equations (Fiedler, B., Gröger, K. and Sprekels, J., eds), World Scientific, pp. 289294.Google Scholar
Baiti, P., LeFloch, P. G. and Piccoli, B. (2001), ‘Uniqueness of classical and nonclassical solutions for nonlinear hyperbolic systems’, J. Differ. Equ. 172, 5982.Google Scholar
Baiti, P., LeFloch, P. G. and Piccoli, B. (2004), ‘Existence theory for nonclassical entropy solutions: Scalar conservation laws’, Z. Angew. Math. Phys. 55, 927945.Google Scholar
Bank, M. and Ben-Artzi, M. (2010), ‘Scalar conservation laws on a half-line: A parabolic approach’, J. Hyperbolic Differ. Equ. 7, 165189.CrossRefGoogle Scholar
Bardos, C. W., Le Roux, A.-Y. and Nedelec, J.-C. (1979), ‘First order quasilinear equations with boundary conditions’, Comm. Part. Diff. Equ. 4, 10181034.Google Scholar
Bedjaoui, N. and LeFloch, P. G. (2002a), ‘Diffusive-dispersive traveling waves and kinetic relations I: Non-convex hyperbolic conservation laws’, J. Differ. Equ. 178, 574607.Google Scholar
Bedjaoui, N. and LeFloch, P. G. (2002b), ‘Diffusive-dispersive traveling waves and kinetic relations II: A hyperbolic–elliptic model of phase transitions dynamics’, Proc. Royal Soc. Edinburgh A 132, 545565.Google Scholar
Bedjaoui, N. and LeFloch, P. G. (2002c), ‘Diffusive–dispersive traveling waves and kinetic relations III: An hyperbolic model from nonlinear elastodynamics’, Ann. Univ. Ferrara Sc. Mat. 47, 117144.Google Scholar
Bedjaoui, N. and LeFloch, P. G. (2003), ‘Diffusive–dispersive traveling waves and kinetic relations IV: Compressible Euler system’, Chinese Ann. Appl. Math. 24, 1734.CrossRefGoogle Scholar
Bedjaoui, N. and LeFloch, P. G. (2004), ‘Diffusive–dispersive traveling waves and kinetic relations V: Singular diffusion and dispersion terms’, Proc. Royal Soc. Edinburgh A 134, 815844.Google Scholar
Bedjaoui, N., Chalons, C., Coquel, F. and LeFloch, P. G. (2005), ‘Non-monotonic traveling waves in van der Waals fluids’, Anal. Appl. 3, 419446.Google Scholar
Beljadid, A., LeFloch, P. G., Mishra, S. and Parés, C. (2014), Schemes with well- controlled dissipation (WCD). In preparation.Google Scholar
Benabdallah, A. (1986), ‘Le p-systeme dans un intervalle’, CR Acad. Sci. Paris, 303, 123126.Google Scholar
Benzoni-Gavage, S. (1999), ‘Stability of subsonic planar phase boundaries in a van der Waals fluid’, Arch. Ration. Mech. Anal. 150, 2355.CrossRefGoogle Scholar
Beretta, E., Hulshof, J. and Peletier, L. A. (1996), ‘On an ordinary differential equation from forced coating flow’, J. Differ. Equ. 130, 247265.Google Scholar
Berthon, C. (2002), ‘Nonlinear scheme for approximating a nonconservative hyperbolic system’, CR Acad. Sci. Paris 335, 10691072.Google Scholar
Berthon, C. and Coquel, F. (1999), Nonlinear projection methods for multi-entropies Navier–Stokes systems. In Finite Volumes for Complex Applications II: Problems and Perspectives, Hermes Science Publications, pp. 307314.Google Scholar
Berthon, C. and Coquel, F. (2002), Nonlinear projection methods for multi-entropies Navier-Stokes systems. In Innovative Methods for Numerical Solutions of Partial Differential Equations, World Scientific, pp. 278304.Google Scholar
Berthon, C. and Coquel, F. (2006), ‘Multiple solutions for compressible turbulent flow models’, Commun. Math. Sci. 4, 497511.CrossRefGoogle Scholar
Berthon, C. and Coquel, F. (2007), ‘Nonlinear projection methods for multi-entropies Navier-Stokes systems’, Math. Comp. 76, 11631194.Google Scholar
Berthon, C. and Foucher, F. (2012), ‘Efficient well-balanced hydrostatic upwind schemes for shallow-water equations’, J. Comput. Phys. 231, 49935015.Google Scholar
Berthon, C., Coquel, F. and LeFloch, P. G. (2002), Kinetic relations for nonconservative hyperbolic systems and applications. Unpublished notes.Google Scholar
Berthon, C., Coquel, F. and LeFloch, P. G. (2012), ‘Why many theories of shock waves are necessary: Kinetic relations for nonconservative systems’, Proc. Royal Soc. Edinburgh 137, 137.CrossRefGoogle Scholar
Bertozzi, A. and Shearer, M. (2000), ‘Existence of undercompressive traveling waves in thin film equations’, SIAM J. Math. Anal. 32, 194213.Google Scholar
Bertozzi, A., Muönch, A. and Shearer, M. (2000), ‘Undercompressive shocks in thin film flow’, Phys. D 134, 431464.Google Scholar
Bertozzi, A., Mönch, A., Shearer, M. and Zumbrun, K. (2001), ‘Stability of compressive and undercompressive thin film traveling waves’, Europ. J. Appl. Math. 12, 253291.Google Scholar
Bianchini, S. and Spinolo, L. V. (2009), ‘The boundary Riemann solver coming from the real vanishing viscosity approximation’, Arch. Ration. Mech. Anal. 191, 196.Google Scholar
Böhme, T., Dreyer, W. and Möller, W. H. (2007), ‘Determination of stiffness and higher gradient coefficients by means of the embedded-atom method’, Contin. Mech. Thermodyn. 18, 411441.Google Scholar
Bouchut, F. and Boyaval, S. (2013) ‘A new model for shallow viscoelastic fluids’, Math. Models Meth. Appl. Sci. 23, 14791526.CrossRefGoogle Scholar
Bouchut, F., Klingenberg, C. and Waagan, K. (2007), ‘A multiwave approximate Riemann solver for ideal MHD based on relaxation I: Theoretical framework’, Numer. Math. 108, 742.Google Scholar
Boutin, B., Chalons, C., Lagoutière, F. and LeFloch, P. G. (2008), ‘Convergent and conservative schemes for nonclassical solutions based on kinetic relations’, Interfaces Free Bound. 10, 399421.Google Scholar
Boutin, B., Coquel, F. and LeFloch, P. G. (2011), ‘Coupling techniques for nonlinear hyperbolic equations I: Self-similar diffusion for thin interfaces’, Proc. Roy. Soc. Edinburgh A 141, 921956.Google Scholar
Boutin, B., Coquel, F. and LeFloch, P. G. (2012), Coupling techniques for nonlinear hyperbolic equations IV: Well-balanced schemes for scalar multidimensional and multi-component laws. Math. Comp., to appear. arXiv:1206.0248Google Scholar
Boutin, B., Coquel, F. and LeFloch, P. G. (2013), ‘Coupling techniques for nonlinear hyperbolic equations III: The well-balanced approximation of thick interfaces’, SIAM J. Numer. Anal. 51, 11081133.CrossRefGoogle Scholar
Bressan, A. and Constantin, A. (2007), ‘Global dissipative solutions of the Camassa–Holm equation’, Anal. Appl. 5, 127.Google Scholar
Brio, M. and Hunter, J. K. (1990), ‘Rotationally invariant hyperbolic waves’, Comm. Pure Appl. Math. 43, 10371053.Google Scholar
Burger, R. and Karlsen, K. H. (2008), ‘Conservation laws with discontinuous flux: A short introduction’, J. Engng Math. 60, 241247.Google Scholar
Caginalp, G. and Chen, X. (1998), ‘Convergence of the phase field model to its sharp interface limits’, Europ. J. Appl. Math. 9, 417445.Google Scholar
Carbou, G. and Hanouzet, B. (2007), ‘Relaxation approximation of some initial-boundary value problem for p-systems’, Commun. Math. Sci. 5, 187203.CrossRefGoogle Scholar
Castro, M. J., Fjordholm, U. S., Mishra, S. and Parés, C. (2013), ‘Entropy conservative and entropy stable schemes for nonconservative hyperbolic systems’, SIAM J. Numer. Anal. 51, 13711391.CrossRefGoogle Scholar
Castro, M. J., Gallardoand, J. M.Parés, C. (2006), ‘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, 11031134.Google Scholar
Castro, M. J., LeFloch, P. G., Muñoz-Ruiz, M. L. and Parés, C. (2008), ‘Why many theories of shock waves are necessary: Convergence error in formally path-consistent schemes’, J. Comput. Phys. 227, 81078129.Google Scholar
Castro, M. J., Macías, J. and Parés, C. (2001), ‘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, 107127.Google Scholar
Castro, M. J., Parés, C., Puppo, G. and Russo, G. (2012), ‘Central schemes for nonconservative hyperbolic systems’, SIAM J. Sci. Comput. B 34, 523558.Google Scholar
Chalons, C. (2008), ‘Transport-equilibrium schemes for computing nonclassical shocks: Scalar conservation laws’, Numer. Methods Partial Differ. Equ. 24, 11271147.Google Scholar
Chalons, C. and Coquel, F. (2007), Numerical capture of shock solutions of nonconservative hyperbolic systems via kinetic functions. In Analysis and Simulation ofFluid Dynamics, Advances in Mathematical Fluid Mechanics, Birkhöauser, pp. 4568.Google Scholar
Chalons, C. and Goatin, P. (2008), ‘Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling’, Interfaces Free Bound. 10, 197221.Google Scholar
Chalons, C. and LeFloch, P. G. (2001a), ‘High-order entropy conservative schemes and kinetic relations for van der Waals fluids’, J. Comput. Phys. 167, 123.Google Scholar
Chalons, C. and LeFloch, P. G. (2001b), ‘A fully discrete scheme for diffusive-dispersive conservation laws’, Numer. Math. 89, 493509.Google Scholar
Chalons, C. and LeFloch, P. G. (2003), ‘Computing undercompressive waves with the random choice scheme: Nonclassical shock waves’, Interfaces Free Bound. 5, 129158.Google Scholar
Chalons, C., Coquel, F., Engel, P. and Rohde, C. (2012), ‘Fast relaxation solvers for hyperbolic-elliptic phase transition problems’, SIAM J. Sci. Comput. A 34, 17531776.Google Scholar
Chalons, C., Raviart, P.-A. and Seguin, N. (2008), ‘The interface coupling of the gas dynamics equations’, Quart. Appl. Math. 66, 659705.Google Scholar
Charlotte, M. and Truskinovsky, L. (2008), ‘Towards multi-scale continuum elasticity theory’, Contin. Mech. Thermodyn. 20, 133161.Google Scholar
Christoforou, C. and Spinolo, L. V. (2011), ‘A uniqueness criterion for viscous limits of boundary Riemann problems’, J. Hyperbolic Differ. Equ. 8, 507544.Google Scholar
Coclite, G. M. and Karlsen, K. H. (2006), ‘A singular limit problem for conservation laws related to the Camassa–Holm shallow water equation’, Comm. Partial Differ. Equ. 31, 12531272.Google Scholar
Colombo, R. M. and Corli, A. (1999), ‘Continuous dependence in conservation laws with phase transitions’, SIAM J. Math. Anal. 31, 3462.Google Scholar
Colombo, R. M. and Corli, A. (2004a), ‘Stability of the Riemann semigroup with respect to the kinetic condition’, Quart. Appl. Math. 62, 541551.Google Scholar
Colombo, R. M. and Corli, A. (2004b), ‘Sonic and kinetic phase transitions with applications to Chapman–Jouguet deflagration’, Math. Methods Appl. Sci. 27, 843864.Google Scholar
Colombo, R. M., Goatin, P. and Piccoli, B. (2010), ‘Road networks with phase transitions’, J. Hyperbolic Differ. Equ. 7, 85106.Google Scholar
Colombo, R., Kröoner, D. and LeFloch, P. G. (2008), Mini-workshop: Hyperbolic aspects of phase transition dynamics. Abstracts from the mini-workshop held February 24-March 1, 2008. Vol. 5, no. 1 of Oberwolfach Reports, European Mathematical Society, pp. 513556.Google Scholar
Coquel, F., Diehl, D., Merkle, C. and Rohde, C. (2005), Sharp and diffuse interface methods for phase transition problems in liquid–vapour flows. In Numerical Methods for Hyperbolic and Kinetic Problems, Vol. 7 of IRMA Lectures in Mathematics and Theoretical Physics, European Mathematical Society, pp. 239270.Google Scholar
Corli, A. and Sablé-Tougeron, M. (1997a), ‘Stability of contact discontinuities under perturbations of bounded variation’, Rend. Sem. Mat. Univ. Padova 97, 3560.Google Scholar
Corli, A. and Sablé-Tougeron, M. (1997b), ‘Perturbations of bounded variation of a strong shock wave’, J. Differ. Equ. 138, 195228.Google Scholar
Corli, A. and Sablé-Tougeron, M. (2000), ‘Kinetic stabilization of a nonlinear sonic phase boundary’, Arch. Ration. Mech. Anal. 152, 163.Google Scholar
Dafermos, C. M. (1972), ‘Polygonal approximations of solutions of the initial value problem for a conservation law’, J. Math. Anal. Appl. 38, 3341.Google Scholar
Dafermos, C. M. (1983), Hyperbolic systems of conservation laws. In Proc. Systems of Nonlinear Partial Differential Equations' (Ball, J. M., ed.), Vol. 111 of NATO ASI series C, Springer, pp. 2570.Google Scholar
Dafermos, C. M. (2000), Hyperbolic Conservation Laws in Continuum Physics, Vol. 325 of Grundlehren der mathematischen Wissenschaft, Springer.Google Scholar
Dal, G. Maso, LeFloch, P. G. and Murat, F. (1990), Definition and weak stability of nonconservative products. Internal report, Centre de Mathématiques Appliquées, Ecole Polytechnique, France.Google Scholar
Dal Maso, G., LeFloch, P. G. and Murat, F. (1995), ‘Definition and weak stability of nonconservative products’, J. Math. Pures Appl. 74, 483548.Google Scholar
De, C. Lellis, Otto, F. and Westdickenberg, M. (2004), ‘Minimal entropy conditions for Burgers equation’, Quart. Appl. Math. 62, 687700.Google Scholar
Donadello, C. and Marson, A. (2007), ‘Stability of front tracking solutions to the initial and boundary value problem for systems of conservation laws’, NoDEA: Nonlinear Differ. Equ. Appl. 14, 569592.Google Scholar
Dreyer, W. and Herrmann, M. (2008), ‘Numerical experiments on the modulation theory for the nonlinear atomic chain’, Phys. D 237, 255282.Google Scholar
Dreyer, W. and Kunik, M. (1998), ‘Maximum entropy principle revisited’, Contin. Mech. Thermodyn. 10, 331347.Google Scholar
Dubois, F. and LeFloch, P. G. (1988), ‘Boundary conditions for nonlinear hyperbolic systems of conservation laws’, J. Differ. Equ. 71, 93122.CrossRefGoogle Scholar
Dutta, R. (2013), Personal communication. University of Oslo, Norway, 05 2013.Google Scholar
E, W. and Li, D. (2008), ‘The Andersen thermostat in molecular dynamics’, Comm. Pure Appl. Math. 61, 96136.Google Scholar
Ernest, J., LeFloch, P. G. and Mishra, S. (2013), Schemes with well-controlled dissipation (WCD). Submitted for publication.Google Scholar
Fan, H. T. and Liu, H.-L. (2004), ‘Pattern formation, wave propagation and stability in conservation laws with slow diffusion and fast reaction’, J. Differ. Hyper. Equ. 1, 605636.Google Scholar
Fan, H.-T. and Slemrod, M. (1993), The Riemann problem for systems of conservation laws of mixed type. In Shock Induced Transitions and Phase Structures in General Media (Dunn, J. E.et al., eds), Vol. 52 of IMA Volumes in Mathematics and its Applications, Springer, pp. 6191.Google Scholar
Fjordholm, U. S. and Mishra, S. (2012), ‘Accurate numerical discretizations of nonconservative hyperbolic systems’, M2AN: Math. Model. Numer. Anal. 46, 187296.CrossRefGoogle Scholar
Fjordholm, U. S., Mishra, S. and Tadmor, E. (2009), Energy preserving and energy stable schemes for the shallow water equations. In Foundations of Computational Mathematics 2008, Vol. 363 of London Mathematical Society Lecture Notes, Cambridge University Press, pp. 93139.Google Scholar
Fjordholm, U. S., Mishra, S. and Tadmor, E. (2011), ‘Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography’, J. Comput. Phys. 230, 55875609.Google Scholar
Fjordholm, U. S., Mishra, S. and Tadmor, E. (2012), ‘Arbitrary order accurate essentially non-oscillatory entropy stable schemes for systems of conservation laws’, SIAM J. Numer. Anal. 50, 544573.Google Scholar
Fjordholm, U. S., Mishra, S. and Tadmor, E. (2013), ‘ENO reconstruction and ENO interpolation are stable’, J. Found. Comp. Math. 13, 139159.Google Scholar
Freistuöhler, H. (1992), ‘Dynamical stability and vanishing viscosity: A case study of a non-strictly hyperbolic system of conservation laws’, Comm. Pure Appl. Math. 45, 561582.Google Scholar
Freistuöhler, H. and Pitman, E. B. (1992), ‘A numerical study of a rotationally degenerate hyperbolic system I: The Riemann problem’, J. Comput. Phys. 100, 306321.Google Scholar
Freistuöhler, H. and Pitman, E. B. (1995), ‘A numerical study of a rotationally degenerate hyperbolic system II: The Cauchy problem’, SIAM J. Numer. Anal. 32, 741753.Google Scholar
Frid, H. and Liu, I.-S. (1996), Phase transitions and oscillation waves in an elastic bar. In Fourth Workshop on Partial Differential Equations, part I, Vol. 11 of Matemática Contemporânea, Sociedade Brasileira de Matemática, pp. 123135.Google Scholar
Gisclon, M. (1996), ‘Étude des conditions aux limites pour un systéme strictement hyperbolique, via l'approximation parabolique’, J. Math. Pures Appl. 9, 485508.Google Scholar
Gisclon, M. and Serre, D. (1994), ‘Étude des conditions aux limites pour un systéme strictement hyperbolique, via l'approximation parabolique’, CR Acad. Sci. Paris 319, 377382.Google Scholar
Goatin, P. and LeFloch, P. G. (2001), ‘Sharp L1 continuous dependence of solutions of bounded variation for hyperbolic systems of conservation laws’, Arch. Ration. Mech. Anal. 157, 3573.Google Scholar
Godlewski, E. and Raviart, P.-A. (2004), ‘The numerical interface coupling of nonlinear hyperbolic systems of conservation laws I: The scalar case’, Numer. Math. 97, 81130.Google Scholar
Godlewski, E., and Le Thanh, K.-C. and Raviart, P.-A. (2005), ‘The numerical interface coupling of nonlinear hyperbolic systems of conservation laws II: The case of systems’, M2AN: Math. Model. Numer. Anal. 39, 649692.Google Scholar
Goodman, J. (1982), PhD thesis, Stanford University, unpublished.Google Scholar
Grinfeld, M. (1989), ‘Nonisothermal dynamic phase transitions’, Quart. Appl. Math. 47, 7184.Google Scholar
Hattori, H. (1986), ‘The Riemann problem for a van der Waals fluid with entropy rate admissibility criterion: isothermal case’, Arch. Ration. Mech. Anal. 92, 247263.Google Scholar
Hattori, H. (2003), ‘The existence and large time behavior of solutions to a system related to a phase transition problem’, SIAM J. Math. Anal. 34, 774804.Google Scholar
Hattori, H. (2008), ‘Existence of solutions with moving phase boundaries in ther-moelasticity’, J. Hyperbolic Differ. Equ. 5, 589611.Google Scholar
Hayes, B. T. and LeFloch, P. G. (1996), Nonclassical shock waves and kinetic relations: Strictly hyperbolic systems. Preprint 357, CMAP, Ecole Polytechnique, Palaiseau, France.Google Scholar
Hayes, B. T. and LeFloch, P. G. (1997), ‘Nonclassical shocks and kinetic relations: Scalar conservation laws’, Arch. Ration. Mech. Anal. 139, 156.Google Scholar
Hayes, B. T. and LeFloch, P. G. (1998), ‘Nonclassical shocks and kinetic relations: Finite difference schemes’, SIAM J. Numer. Anal. 35, 21692194.CrossRefGoogle Scholar
Hayes, B. T. and LeFloch, P. G. (2000), ‘Nonclassical shock waves and kinetic relations: Strictly hyperbolic systems’, SIAM J. Math. Anal. 31, 941991.Google Scholar
Hayes, B. T. and Shearer, M. (1999), ‘Undercompressive shocks and Riemann problems for scalar conservation laws with non-convex fluxes’, Proc. Roy. Soc. Edinburgh A 129, 733754.Google Scholar
Hiltebrand, A. and Mishra, S. (2014), ‘Entropy stable shock capturing streamline diffusion space-time discontinuous Galerkin (DG) methods for systems of conservation laws’, Numer. Math. 126, 103151.Google Scholar
Holden, H., Karlsen, K. H., Mitrovic, D. and Panov, E. Y. (2009), ‘Strong compactness of approximate solutions to degenerate elliptic-hyperbolic equations with discontinuous flux function’, Acta Math. Sci. B 29, 15731612.Google Scholar
Holden, H., Risebro, N. and Sande, H. (2009), ‘The solution of the Cauchy problem with large data for a model of a mixture of gases’, J. Hyperbolic Differ. Equ. 6, 25106.Google Scholar
Hou, T. Y. and LeFloch, P. G. (1994), ‘Why nonconservative schemes converge to wrong solutions: Error analysis’, Math. Comp. 62, 497530.Google Scholar
Hou, T. Y., Rosakis, P. and LeFloch, P. G. (1999), ‘A level set approach to the computation of twinning and phase transition dynamics’, J. Comput. Phys. 150, 302331.Google Scholar
Hu, J. and LeFloch, P. G. (2000), ‘L1 continuous dependence property for systems of conservation laws’, Arch. Ration. Mech. Anal. 151, 4593.Google Scholar
Hwang, S. (2004), ‘Kinetic decomposition for the generalized BBM-Burgers equations with dissipative term’, Proc. Roy. Soc. Edinburgh A 134, 11491162.CrossRefGoogle Scholar
Hwang, S. (2007), ‘Singular limit problem of the Camassa–Holm type equation’, J. Differ. Equ. 235, 7484.Google Scholar
Hwang, S. and Tzavaras, A. (2002), ‘Kinetic decomposition of approximate solutions to conservation laws: Application to relaxation and diffusion–dispersion approximations’, Comm. Partial Differ. Equ. 27, 12291254.Google Scholar
Iguchi, T. and LeFloch, P. G. (2003), ‘Existence theory for hyperbolic systems of conservation laws with general flux-functions’, Arch. Ration. Mech. Anal. 168, 165244.Google Scholar
Isaacson, E., Marchesin, D., Palmeira, C. F. and Plohr, B. J. (1992), ‘A global formalism for nonlinear waves in conservation laws’, Comm. Math. Phys. 146, 505552.Google Scholar
Ismail, F. and Roe, P. L. (2009), ‘Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks’, J. Comput. Phys. 228, 54105436.Google Scholar
Jacobs, D., McKinney, W. R. and Shearer, M. (1995), ‘Traveling wave solutions ofthe modified Korteweg–de Vries Burgers equation’, J. Differ. Equ. 116, 448467.Google Scholar
Jerome, J. W. (1994), The mathematical study and approximation of semiconductor models. In Large-Scale Matrix Problems and the Numerical Solution of Partial Differential Equations, Vol. 3 of Advances in Numerical Analysis, Oxford University Press, pp. 157204,Google Scholar
Joseph, K. T. and LeFloch, P. G. (1996), Boundary layers in weak solutions to hyperbolic conservation laws. Preprint 341, Centre de Mathematiques Appliquees, Ecole Polytechnique, France.Google Scholar
Joseph, K. T. and LeFloch, P. G. (1999), ‘Boundary layers in weak solutions to hyperbolic conservation laws’, Arch. Ration. Mech Anal. 147, 4788.Google Scholar
Joseph, K. T. and LeFloch, P. G. (2002), ‘Boundary layers in weak solutions of hyperbolic conservation laws II: Self-similar vanishing diffusion limits’ (English summary), Commun. Pure Appl. Anal. 1, 5176.Google Scholar
Joseph, K. T. and LeFloch, P. G. (2006), Singular limits for the Riemann problem: General diffusion, relaxation, and boundary conditions. In New Analytical Approach to Multidimensional Balance Laws (Rozanova, O., ed.), Nova, pp. 143172.Google Scholar
Joseph, K. T. and LeFloch, P. G. (2007a), ‘Singular limits in phase dynamics with physical viscosity and capillarity’, Proc. Royal Soc. Edinburgh A 137, 12871312.Google Scholar
Joseph, K. T. and LeFloch, P. G. (2007b), ‘Singular limits for the Riemann problem: general diffusion, relaxation, and boundary conditions’, CR Math. Acad. Sci. Paris 344, 5964.Google Scholar
Karlsen, K. H., Lie, K. and Risebro, N. H. (1999), A front tracking method for conservation laws with boundary conditions. In Hyperbolic Problems: Theory, Numerics, Applications I, Vol. 129 of International Series of Numerical Mathematics, Birkhöauser, pp. 493502.Google Scholar
Karni, S. (1992), ‘Viscous shock profiles and primitive formulations’, SIAM J. Numer. Anal. 29, 15921609.Google Scholar
Kissling, F. and Rohde, C. (2010), ‘The computation of nonclassical shock waves with a heterogeneous multiscale method’, Netw. Heterog. Media 5, 661674.Google Scholar
Kissling, F., LeFloch, P. G. and Rohde, C. (2009), ‘Singular limits and kinetic decomposition for a non-local diffusion-dispersion problem’, J. Differ. Equ. 247, 33383356.Google Scholar
Kondo, C. and LeFloch, P. G. (2002), ‘Zero diffusion-dispersion limits for hyperbolic conservation laws’, SIAM Math. Anal. 33, 13201329.Google Scholar
Kröoner, D., LeFloch, P. G. and Thanh, M. D. (2008), ‘The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section’, M2AN: Math. Model. Numer. Anal. 42, 425442.Google Scholar
Kruzkov, S. N. (1970), ‘First order quasilinear equations with several independent variables’, Mat. Sb. (N.S.) 81, 228–25.Google Scholar
Kumar, H. and Mishra, S. (2012), ‘Entropy stable numerical schemes for two-fluid MHD equations’, J. Sci. Comput. 52, 401425.Google Scholar
Kwon, Y.-S. (2009), ‘Diffusion-dispersion limits for multidimensional scalar conservation laws with source terms’, J. Differ. Equ. 246, 18831893.Google Scholar
Laforest, M. and LeFloch, P. G. (2010), ‘Diminishing functionals for nonclassical entropy solutions selected by kinetic relations’, Port. Math. 67, 279319.Google Scholar
Laforest, M. and LeFloch, P. G. (2014), in preparation.Google Scholar
Lax, P. D. (1957), ‘Hyperbolic systems of conservation laws II’, Comm. Pure Appl. Math. 10, 537566.Google Scholar
Lax, P. D. (1973), Hyperbolic Systems ofConservation Laws and the Mathematical Theory of Shock Waves, Vol.11 of Regional Confer. Series in Appl. Math., SIAM.Google Scholar
Lax, P. D. and Levermore, C. D. (1983), ‘The small dispersion limit of the Korteweg–de Vries equation’, Comm. Pure Appl. Math. 36, 253290.Google Scholar
Lax, P. D. and Wendroff, B. (1960), ‘Systems of conservation laws’, Comm. Pure Appl. Math. 13, 217237.Google Scholar
Lax, P. D. and Wendroff, B. (1962), ‘On the stability of difference schemes’, Comm. Pure Appl. Math. 15, 363371.Google Scholar
LeFloch, P. G. (1988), ‘Entropy weak solutions to nonlinear hyperbolic systems in nonconservative form’, Comm. Part. Diff. Equ. 13, 669727.Google Scholar
LeFloch, P. G. (1989), Shock waves for nonlinear hyperbolic systems in nonconservative form. IMA preprint 593.Google Scholar
LeFloch, P. G. (1990), An existence and uniqueness result for two nonstrictly hyperbolic systems. In Nonlinear Evolution Equations That Change Type, Vol. 27 of IMA Volumes in Mathematics and its Applications, Springer, pp. 126138.Google Scholar
LeFloch, P. G. (1993), ‘Propagating phase boundaries: Formulation of the problem and existence via the Glimm scheme’, Arch. Ration. Mech. Anal. 123, 153197.Google Scholar
LeFloch, P. G. (1999), An introduction to nonclassical shocks of systems of conservation laws. In An Introduction to Recent Developments in Theory and Numerics for Conservation Laws (Kroöner, D., Ohlberger, M. and Rohde, C., eds), Vol. 5 of Lecture Notes in Computational Science and Engineering, Springer, pp. 2872.Google Scholar
LeFloch, P. G. (2002), Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, ETH Zurich Lectures in Mathematics, Birkhäuser.Google Scholar
LeFloch, P. G. (2004), ‘Graph solutions of nonlinear hyperbolic systems’, J. Hyperbolic Differ. Equ. 1, 643689.Google Scholar
LeFloch, P. G. (2010), 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, pp. 237272.Google Scholar
LeFloch, P. G. and Liu, T.-P. (1993), ‘Existence theory for nonlinear hyperbolic systems in nonconservative form’, Forum Math. 5, 261280.Google Scholar
LeFloch, P. G. and Mishra, S. (2009), ‘Nonclassical shocks and numerical kinetic relations for a model MHD system’, Act. Math. Sci. 29, 16841702.Google Scholar
LeFloch, P. G. and Mohamadian, M. (2008), ‘Why many shock wave theories are necessary: Fourth-order models, kinetic functions, and equivalent equations’, J. Comput. Phys. 227, 41624189.Google Scholar
LeFloch, P. G. and Natalini, R. (1999), ‘Conservation laws with vanishing nonlinear diffusion and dispersion’, Nonlinear Analysis 36, 213230.Google Scholar
LeFloch, P. G. and Rohde, C. (2000), ‘High-order schemes, entropy inequalities, and nonclassical shocks’, SIAM J. Numer. Anal. 37, 20232060.Google Scholar
LeFloch, P. G. and Shearer, M. (2004), ‘Nonclassical Riemann solvers with nucleation’, Proc. Royal Soc. Edinburgh A 134, 941964.Google Scholar
LeFloch, P. G. and Thanh, M. D. (2000), ‘Nonclassical Riemann solvers and kinetic relations III: A nonconvex hyperbolic model for van der Waals fluids’, Electr. J. Differ. Equ. 72, 119.Google Scholar
LeFloch, P. G. and Thanh, M. D. (2001a), ‘Nonclassical Riemann solvers and kinetic relations I: A nonconvex hyperbolic model of phase transitions’, Z. Angew. Math. Phys. 52, 597619.Google Scholar
LeFloch, P. G. and Thanh, M. D. (2001b), ‘Nonclassical Riemann solvers and kinetic relations II: An hyperbolic-elliptic model of phase transitions’, Proc. Royal Soc. Edinburgh A 132, 181219.Google Scholar
LeFloch, P. G. and Thanh, M. D. (2003), ‘Properties of Rankine-Hugoniot curves for van der Waals fluids’, Japan J. Indust. Applied Math. 20, 211238.Google Scholar
LeFloch, P. G. and Thanh, M. D. (2007), ‘The Riemann problem for the shallow water equations with discontinuous topography’, Commun. Math. Sci. 5, 865885.Google Scholar
LeFloch, P. G. and Thanh, M. D. (2011), ‘A Godunov-type method for the shallow water equations with discontinuous topography in the resonant regime’, J. Comput. Phys. 230, 76317660.Google Scholar
LeFloch, P. G., Mercier, J.-M. and Rohde, C. (2002), ‘Fully discrete entropy conservative schemes of arbitrary order’, SIAM J. Numer. Anal. 40, 19681992.Google Scholar
Le, A.-Y. Roux (1977), ‘Étude du probleme mixte pour une equation quasi-lineaire du premier ordre’ (in French), CR Acad. Sci. Paris A/B 285, 351354.Google Scholar
LeVeque, R. J. (2003), Finite Volume Methods for Hyperbolic Problems, Cambridge University Press.Google Scholar
Levy, R. and Shearer, M. (2004), ‘Comparison of two dynamic contact line models for driven thin liquid films’, Europ. J. Appl. Math. 15, 625642.Google Scholar
Levy, R. and Shearer, M. (2005), ‘Kinetics and nucleation for driven thin film flow’, Phys. D 209, 145163.Google Scholar
Li, H.-L. and Pan, R.-H. (2000), ‘Zero relaxation limit for piecewise smooth solutions to a rate-type viscoelastic system in the presence of shocks’, J. Math. Anal. Appl. 252, 298324.Google Scholar
Mercier, J.-M. and Piccoli, B. (2001), ‘Global continuous Riemann solver for nonlinear elasticity’, Arch. Ration. Mech. Anal. 156, 89119.Google Scholar
Mercier, J.-M. and Piccoli, B. (2002), ‘Admissible Riemann solvers for genuinely nonlinear p-systems of mixed type’, J. Differ. Equ. 180, 395426.Google Scholar
Merkle, C. and Rohde, C. (2006), ‘Computation of dynamical phase transitions in solids’, Appl. Numer. Math. 56, 14501463.Google Scholar
Merkle, C. and Rohde, C. (2007), ‘The sharp-interface approach for fluids with phase change: Riemann problems and ghost fluid techniques’, M2AN: Math. Model. Numer. Anal. 41, 10891123.Google Scholar
Mishra, S. and L.Spinolo, V. (2011), Accurate numerical schemes for approximating initial-boundary value problems for systems of conservation laws. Preprint 2011, SAM report 2011-57, ETH Zürich.Google Scholar
Morando, A. and Secchi, P. (2011), ‘Regularity of weakly well posed hyperbolic mixed problems with characteristic boundary’, J. Hyperbolic Differ. Equ. 8, 3799.Google Scholar
Morin, A., Flatten, T. and Munkejord, S. T. (2013), ‘A Roe scheme for a compressible six-equation two-fluid model’, Internat. J. Numer. Methods Fluids 72, 478504.Google Scholar
Münch, A. (2000), ‘Shock transitions in Marangoni gravity-driven thin-film flow’, Nonlinearity 13, 731746.Google Scholar
Muñoz, M. L. and Parés, C. (2007), ‘Godunov method for nonconservative hyperbolic systems’, Math. Model. Numer. Anal. 41, 169185.Google Scholar
Murillo, J. and García-Navarro, P. (2013), ‘Energy balance numerical schemes for shallow water equations with discontinuous topography’, J. Comput. Phys. 236, 119142.Google Scholar
Ngan, S.-C. and Truskinovsky, L. (2002), ‘Thermo-elastic aspects of dynamic nucleation’, J. Mech. Phys. Solids 50, 11931229.Google Scholar
Oleinik, O. (1963), ‘Discontinuous solutions of nonlinear differential equations’, Trans. Amer. Math. Soc. 26, 95172.Google Scholar
Otto, F. and Westdickenberg, M. (2005), ‘Convergence of thin film approximation for a scalar conservation law’, J. Hyperbolic Differ. Equ. 2, 183199.Google Scholar
Parés, C. (2006), ‘Numerical methods for nonconservative hyperbolic systems: A theoretical framework’, SIAM J. Numer. Anal. 44, 300321.Google Scholar
Rütz, A. and Voigt, A. (2007), ‘A diffuse-interface approximation for surface diffusion including adatoms’, Nonlinearity 20, 177192.Google Scholar
Rohde, C. (2005a), ‘Scalar conservation laws with mixed local and non-local diffusion-dispersion terms’, SIAM J. Math. Anal. 37, 103129.Google Scholar
Rohde, C. (2005b), ‘On local and non-local Navier–Stokes-Korteweg systems for liquid-vapour phase transitions’, Z. Angew. Math. Mech. 85, 839857.Google Scholar
Saurel, R. and Abgrall, R. (1999), ‘A multiphase Godunov method for compressible multifluid and multiphase flows’, J. Comput. Phys. 150, 425467Google Scholar
Schecter, S. and Shearer, M. (1991), ‘Undercompressive shocks for nonstrictly hyperbolic conservation laws’, Dynamics Diff. Equ. 3, 199271.Google Scholar
Schonbek, M. E. (1982), ‘Convergence of solutions to nonlinear dispersive equations’, Comm. Part. Diff. Equ. 7, 9591000.Google Scholar
Schulze, S. and Shearer, M. (1999), ‘Undercompressive shocks for a system of hyperbolic conservation laws with cubic nonlinearity’, J. Math. Anal. Appl. 229, 344362.Google Scholar
Seguin, N. and Vovelle, J. (2003), ‘Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients’, Math. Models Methods Appl. Sci. 13, 221257.Google Scholar
Serre, D. (2000), Systems of Conservation Laws 2: Geometric Structures, Oscillations, and Initial and Boundary Value Problems, Cambridge University Press.Google Scholar
Serre, D. (2007), Discrete shock profiles: Existence and stability. In Hyperbolic Systems of Balance Laws, Vol. 1911 of Lecture Notes in Mathematics, Springer, pp. 79158.Google Scholar
Shearer, M. (1986), ‘The Riemann problem for the planar motion of an elastic string’, J. Differ. Equ. 61, 149163.Google Scholar
Shearer, M. (1998), ‘Dynamic phase transitions in a van der Waals gas’, Quart. Appl. Math. 46, 631636.Google Scholar
Shearer, M. and Yang, Y. (1995), ‘The Riemann problem for a system of conservation laws of mixed type with a cubic nonlinearity’, Proc. Roy. Soc. Edinburgh A 125, 675699.CrossRefGoogle Scholar
Slemrod, M. (1983), ‘Admissibility criteria for propagating phase boundaries in a van der Waals fluid’, Arch. Ration. Mech. Anal. 81, 301315.Google Scholar
Slemrod, M. (1989), ‘A limiting viscosity approach to the Riemann problem for materials exhibiting change of phase’, Arch. Ration. Mech. Anal. 105, 327365.Google Scholar
Suliciu, I. (1990), ‘On modelling phase transitions by means of rate-type constitutive equations, shock wave structure’, Internat. J. Engng Sci. 28, 829841.Google Scholar
Tadmor, E. (1987), ‘The numerical viscosity of entropy stable schemes for systems of conservation laws I’, Math. Comp. 49, 91103.Google Scholar
Tadmor, E. (2003), ‘Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems’, Acta Numer. 12, 451512.Google Scholar
Tadmor, E. and Zhong, W. (2006), ‘Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity’, J. Hyperbolic Differ. Equ. 3, 529559.Google Scholar
Thanh, M. D. and Krüner, D. (2012), ‘Numerical treatment of nonconservative terms in resonant regime for fluid flows in a nozzle with variable cross-section’, Comput. Fluids 66, 130139.Google Scholar
Truskinovsky, L. (1987), ‘Dynamics of non-equilibrium phase boundaries in a heat conducting non-linearly elastic medium’, J. Appl. Math. Mech. (PMM) 51, 777784.Google Scholar
Truskinovsky, L. (1993), Kinks versus shocks. In Shock Induced Transitions and Phase Structures in General Media (Fosdick, R., Dunn, E. and Slemrod, M., eds), Vol. 52 of IMA Volumes in Mathematics and its Applications, Springer, pp. 185229.Google Scholar
Truskinovsky, L. (1994), ‘Transition to detonation in dynamic phase changes’, Arch. Ration. Mech. Anal. 125, 375397.Google Scholar
Truskinovsky, L. and Vainchtein, A. (2006), ‘Quasicontinuum models of dynamic phase transitions’, Contin. Mech. Thermodyn. 18, 121.Google Scholar
Van Duijn, C. J., Peletier, L. A. and Pop, I. S. (2007), ‘A new class of entropy solutions of the Buckley-Leverett equation’, SIAM J. Math. Anal. 39, 507536.Google Scholar
Volpert, A. I. (1967), ‘The space BV and quasi-linear equations’, Mat. USSR Sb. 2, 225267.Google Scholar
Wendroff, B. (1972), ‘The Riemann problem for materials with non-convex equations of state I: Isentropic flow’, J. Math. Anal. Appl. 38, 454466.Google Scholar
Zhong, X.-G., Hou, T. Y. and LeFloch, P. G. (1996), ‘Computational methods for propagating phase boundaries’, J. Comput. Phys. 124, 192216.Google Scholar