June  2021, 16(2): 283-315. doi: 10.3934/nhm.2021007

Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure

1. 

Univ. Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

2. 

Centre de Mathématique Appliquées, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France

3. 

Laboratoire Jacques-Louis Lions & Centre National de la Recherche Scientifique, Sorbonne Université, 75258 Paris, France

* Corresponding author: Benjamin Boutin

Received  October 2020 Revised  January 2021 Published  June 2021 Early access  February 2021

Fund Project: The three authors were partially supported by the Innovative Training Networks (ITN) grant 642768 (ModCompShock), and by the Centre National de la Recherche Scientifique (CNRS)

In the first part of this series, an augmented PDE system was introduced in order to couple two nonlinear hyperbolic equations together. This formulation allowed the authors, based on Dafermos's self-similar viscosity method, to establish the existence of self-similar solutions to the coupled Riemann problem. We continue here this analysis in the restricted case of one-dimensional scalar equations and investigate the internal structure of the interface in order to derive a selection criterion associated with the underlying regularization mechanism and, in turn, to characterize the nonconservative interface layer. In addition, we identify a new criterion that selects double-waved solutions that are also continuous at the interface. We conclude by providing some evidence that such solutions can be non-unique when dealing with non-convex flux-functions.

Citation: Benjamin Boutin, Frédéric Coquel, Philippe G. LeFloch. Coupling techniques for nonlinear hyperbolic equations. Ⅱ. resonant interfaces with internal structure. Networks and Heterogeneous Media, 2021, 16 (2) : 283-315. doi: 10.3934/nhm.2021007
References:
[1]

Ad imurthiS. Mishra and G. D. V. Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), 783-837.  doi: 10.1142/S0219891605000622.

[2]

Ad imurthiS. Mishra and G. D. V. Gowda, Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes, Netw. Heterog. Media, 2 (2007), 127-157.  doi: 10.3934/nhm.2007.2.127.

[3]

Ad imurthiS. Mishra and G. D. V. Gowda, Conservation law with the flux function discontinuous in the space variable. Ⅱ. Convex-concave type fluxes and generalized entropy solutions, J. Comput. Appl. Math., 203 (2007), 310-344.  doi: 10.1016/j.cam.2006.04.009.

[4]

A. AmbrosoC. ChalonsF. Coquel and T. Galié, Interface model coupling via prescribed local flux balance, ESAIM Math. Model. Numer. Anal., 48 (2014), 895-918.  doi: 10.1051/m2an/2013125.

[5]

B. Andreianov, Dissipative coupling of scalar conservation laws across an interface: theory and applications, in Hyperbolic Problems: Theory, Numerics and Applications, Vol. 17 of Ser. Contemp. Appl. Math. CAM, World Sci. Publishing, Singapore, 1 (2012), 123–135. doi: 10.1142/9789814417099_0009.

[6]

B. Andreianov, The semigroup approach to conservation laws with discontinuous flux, in Hyperbolic Conservation Laws and Related Analysis with Applications, Vol. 49 of Springer Proc. Math. Stat., Springer, Berlin, Heidelberg, (2014), 1–22. doi: 10.1007/978-3-642-39007-4_1.

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B. Andreianov, New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux, in CANUM 2014 - 42e Congrès National d'Analyse Numérique Vol. 50 of ESAIM Proc. Surveys, EDP Sci., Les Ulis, (2015), 40–65. doi: 10.1051/proc/201550003.

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B. Andreianov and C. Cancès, On interface transmission conditions for conservation laws with discontinuous flux of general shape, J. Hyperbolic Differ. Equ., 12 (2015), 343-384.  doi: 10.1142/S0219891615500101.

[9]

B. AndreianovK. H. Karlsen and N. H. Risebro, On vanishing viscosity approximation of conservation laws with discontinuous flux, Netw. Heterog. Media, 5 (2010), 617-633.  doi: 10.3934/nhm.2010.5.617.

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B. AndreianovK. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86.  doi: 10.1007/s00205-010-0389-4.

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B. Andreianov and N. Seguin, Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes, Discrete Contin. Dyn. Syst., 32 (2012), 1939-1964.  doi: 10.3934/dcds.2012.32.1939.

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E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253-265.  doi: 10.1017/S0308210500003863.

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B. BoutinF. Coquel and P. G. LeFloch, Coupling techniques for nonlinear hyperbolic equations. Ⅰ: Self-similar diffusion for thin interfaces, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 921-956.  doi: 10.1017/S0308210510001459.

[20]

B. BoutinF. Coquel and P. G. LeFloch, Coupling techniques for nonlinear hyperbolic equations. Ⅲ. The well–balanced approximation of thick interfaces, SIAM J. Numer. Anal., 51 (2013), 1108-1133.  doi: 10.1137/120865768.

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B. BoutinF. Coquel and P. G. LeFloch, Coupling techniques for nonlinear hyperbolic equations. Ⅳ. Well-balanced schemes for scalar multidimensional and multi-component laws, Math. Comp., 84 (2015), 1663-1702.  doi: 10.1090/S0025-5718-2015-02933-0.

[22]

R. Bürger and K. H. Karlsen, Conservation laws with discontinuous flux: A short introduction, J. Engrg. Math., 60 (2008), 241-247.  doi: 10.1007/s10665-008-9213-7.

[23]

M. J. CastroP. G. LeFlochM. L. Muñoz Ruiz and C. Parés, Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes, J. Comput. Phys., 227 (2008), 8107-8129.  doi: 10.1016/j.jcp.2008.05.012.

[24]

C. Chalons, Theoretical and numerical aspects of the interfacial coupling: the scalar Riemann problem and an application to multiphase flows, Netw. Heterog. Media, 5 (2010), 507-524.  doi: 10.3934/nhm.2010.5.507.

[25]

C. ChalonsP.-A. Raviart and N. Seguin, The interface coupling of the gas dynamics equations, Quart. Appl. Math., 66 (2008), 659-705.  doi: 10.1090/S0033-569X-08-01087-X.

[26]

C. Christoforou and L. V. Spinolo, On the physical and the self-similar viscous approximation of a boundary Riemann problem, Riv. Math. Univ. Parma (N.S.), 3 (2012), 41-54. 

[27]

F. CoquelE. GodlewskiK. HaddaouiC. Marmignon and F. Renac, Choice of measure source terms in interface coupling for a model problem in gas dynamics, Math. Comp., 85 (2016), 2305-2339.  doi: 10.1090/mcom/3063.

[28]

A. CorliM. FigielA. Futa and M. D. Rosini, Coupling conditions for isothermal gas flow and applications to valves, Nonlinear Anal. Real World Appl., 40 (2018), 403-427.  doi: 10.1016/j.nonrwa.2017.09.005.

[29]

C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), 33-41.  doi: 10.1016/0022-247X(72)90114-X.

[30]

C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Arch. Rational Mech. Anal., 52 (1973), 1-9.  doi: 10.1007/BF00249087.

[31]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, fourth edition, 2016. doi: 10.1007/978-3-662-49451-6.

[32]

G. Dal MasoP. G. LeFloch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl., 74 (1995), 483-548. 

[33]

S. Diehl, Scalar conservation laws with discontinuous flux function.Ⅰ. The viscous profile condition, Comm. Math. Phys., 176 (1996), 23-44.  doi: 10.1007/BF02099361.

[34]

S. Diehl and N.-O. Wallin, Scalar conservation laws with discontinuous flux function. Ⅱ. On the stability of the viscous profiles, Comm. Math. Phys., 176 (1996), 45-71.  doi: 10.1007/BF02099362.

[35]

F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 71 (1988), 93-122.  doi: 10.1016/0022-0396(88)90040-X.

[36]

T. Galié, Interface Model Coupling in Fluid Dynamics, Application to Two-Phase Flows, Ph.D thesis, Université Pierre et Marie Curie - Paris VI, available from: https://tel.archives-ouvertes.fr/tel-00395593/, 2009.

[37]

M. Garavello and P. Goatin, The Aw-Rascle traffic model with locally constrained flow, J. Math. Anal. Appl., 378 (2011), 634-648.  doi: 10.1016/j.jmaa.2011.01.033.

[38]

M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model, Comm. Partial Differential Equations, 31 (2006), 243-275.  doi: 10.1080/03605300500358053.

[39]

P. Goatin and P. G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 881–902. doi: 10.1016/j.anihpc.2004.02.002.

[40]

E. GodlewskiK.-C. Le Thanh and P.-A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. Ⅱ. The case of systems, M2AN Math. Model. Numer. Anal., 39 (2005), 649-692.  doi: 10.1051/m2an:2005029.

[41]

E. Godlewski and P.-A. Raviart, Hyperbolic Systems of Conservation Laws, Vol. 3/4 of Mathématiques & Applications, Ellipses, Paris, 1991.

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E. Godlewski and P.-A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. Ⅰ. The scalar case, Numer. Math., 97 (2004), 81-130.  doi: 10.1007/s00211-002-0438-5.

[43]

M. Herty, Modeling, simulation and optimization of gas networks with compressors, Netw. Heterog. Media, 2 (2007), 81-97.  doi: 10.3934/nhm.2007.2.81.

[44]

T. Y. Hou and P. G. LeFloch, Why nonconservative schemes converge to wrong solutions. Error analysis, Math. of Comput., 62 (1994), 497-530.  doi: 10.1090/S0025-5718-1994-1201068-0.

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E. Isaacson and B. Temple, Nonlinear resonance in systems of conservation laws, SIAM J. Appl. Math., 52 (1992), 1260-1278.  doi: 10.1137/0152073.

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K. T. Joseph and P. G. LeFloch, Boundary layers in weak solutions to hyperbolic conservation laws, Arch. Rational Mech Anal., 147 (1999), 47-88.  doi: 10.1007/s002050050145.

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K. T. Joseph and P. G. LeFloch, Boundary layers in weak solutions of hyperbolic conservation laws. Ⅱ. Self-similar vanishing diffusion limits, Commun. Pure Appl. Anal., 1 (2002), 51-76.  doi: 10.3934/cpaa.2002.1.51.

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show all references

References:
[1]

Ad imurthiS. Mishra and G. D. V. Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), 783-837.  doi: 10.1142/S0219891605000622.

[2]

Ad imurthiS. Mishra and G. D. V. Gowda, Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes, Netw. Heterog. Media, 2 (2007), 127-157.  doi: 10.3934/nhm.2007.2.127.

[3]

Ad imurthiS. Mishra and G. D. V. Gowda, Conservation law with the flux function discontinuous in the space variable. Ⅱ. Convex-concave type fluxes and generalized entropy solutions, J. Comput. Appl. Math., 203 (2007), 310-344.  doi: 10.1016/j.cam.2006.04.009.

[4]

A. AmbrosoC. ChalonsF. Coquel and T. Galié, Interface model coupling via prescribed local flux balance, ESAIM Math. Model. Numer. Anal., 48 (2014), 895-918.  doi: 10.1051/m2an/2013125.

[5]

B. Andreianov, Dissipative coupling of scalar conservation laws across an interface: theory and applications, in Hyperbolic Problems: Theory, Numerics and Applications, Vol. 17 of Ser. Contemp. Appl. Math. CAM, World Sci. Publishing, Singapore, 1 (2012), 123–135. doi: 10.1142/9789814417099_0009.

[6]

B. Andreianov, The semigroup approach to conservation laws with discontinuous flux, in Hyperbolic Conservation Laws and Related Analysis with Applications, Vol. 49 of Springer Proc. Math. Stat., Springer, Berlin, Heidelberg, (2014), 1–22. doi: 10.1007/978-3-642-39007-4_1.

[7]

B. Andreianov, New approaches to describing admissibility of solutions of scalar conservation laws with discontinuous flux, in CANUM 2014 - 42e Congrès National d'Analyse Numérique Vol. 50 of ESAIM Proc. Surveys, EDP Sci., Les Ulis, (2015), 40–65. doi: 10.1051/proc/201550003.

[8]

B. Andreianov and C. Cancès, On interface transmission conditions for conservation laws with discontinuous flux of general shape, J. Hyperbolic Differ. Equ., 12 (2015), 343-384.  doi: 10.1142/S0219891615500101.

[9]

B. AndreianovK. H. Karlsen and N. H. Risebro, On vanishing viscosity approximation of conservation laws with discontinuous flux, Netw. Heterog. Media, 5 (2010), 617-633.  doi: 10.3934/nhm.2010.5.617.

[10]

B. AndreianovK. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86.  doi: 10.1007/s00205-010-0389-4.

[11]

B. Andreianov and D. Mitrović, Entropy conditions for scalar conservation laws with discontinuous flux revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), 1307–1335. doi: 10.1016/j.anihpc.2014.08.002.

[12]

B. Andreianov and N. Seguin, Analysis of a Burgers equation with singular resonant source term and convergence of well-balanced schemes, Discrete Contin. Dyn. Syst., 32 (2012), 1939-1964.  doi: 10.3934/dcds.2012.32.1939.

[13]

E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253-265.  doi: 10.1017/S0308210500003863.

[14]

C. BardosA. Y. le Roux and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations, 4 (1979), 1017-1034.  doi: 10.1080/03605307908820117.

[15]

M. BenyahiaC. DonadelloN. Dymski and M. D. Rosini, An existence result for a constrained two-phase transition model with metastable phase for vehicular traffic, NoDEA Nonlinear Differential Equations Appl., 25 (2018), 48-90.  doi: 10.1007/s00030-018-0539-1.

[16]

C. BerthonM. Bessemoulin-ChatardA. Crestetto and F. Foucher, A Riemann solution approximation based on the zero diffusion-dispersion limit of Dafermos reformulation type problem, Calcolo, 56 (2019), 28-60.  doi: 10.1007/s10092-019-0325-4.

[17]

C. BerthonF. Coquel and P. G. LeFloch, Why many theories of shock waves are necessary: Kinetic relations for non-conservative systems, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1-37.  doi: 10.1017/S0308210510001009.

[18]

B. BoutinC. Chalons and P.-A. Raviart, Existence result for the coupling problem of two scalar conservation laws with Riemann initial data, Math. Models Methods Appl. Sci., 20 (2010), 1859-1898.  doi: 10.1142/S0218202510004817.

[19]

B. BoutinF. Coquel and P. G. LeFloch, Coupling techniques for nonlinear hyperbolic equations. Ⅰ: Self-similar diffusion for thin interfaces, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 921-956.  doi: 10.1017/S0308210510001459.

[20]

B. BoutinF. Coquel and P. G. LeFloch, Coupling techniques for nonlinear hyperbolic equations. Ⅲ. The well–balanced approximation of thick interfaces, SIAM J. Numer. Anal., 51 (2013), 1108-1133.  doi: 10.1137/120865768.

[21]

B. BoutinF. Coquel and P. G. LeFloch, Coupling techniques for nonlinear hyperbolic equations. Ⅳ. Well-balanced schemes for scalar multidimensional and multi-component laws, Math. Comp., 84 (2015), 1663-1702.  doi: 10.1090/S0025-5718-2015-02933-0.

[22]

R. Bürger and K. H. Karlsen, Conservation laws with discontinuous flux: A short introduction, J. Engrg. Math., 60 (2008), 241-247.  doi: 10.1007/s10665-008-9213-7.

[23]

M. J. CastroP. G. LeFlochM. L. Muñoz Ruiz and C. Parés, Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes, J. Comput. Phys., 227 (2008), 8107-8129.  doi: 10.1016/j.jcp.2008.05.012.

[24]

C. Chalons, Theoretical and numerical aspects of the interfacial coupling: the scalar Riemann problem and an application to multiphase flows, Netw. Heterog. Media, 5 (2010), 507-524.  doi: 10.3934/nhm.2010.5.507.

[25]

C. ChalonsP.-A. Raviart and N. Seguin, The interface coupling of the gas dynamics equations, Quart. Appl. Math., 66 (2008), 659-705.  doi: 10.1090/S0033-569X-08-01087-X.

[26]

C. Christoforou and L. V. Spinolo, On the physical and the self-similar viscous approximation of a boundary Riemann problem, Riv. Math. Univ. Parma (N.S.), 3 (2012), 41-54. 

[27]

F. CoquelE. GodlewskiK. HaddaouiC. Marmignon and F. Renac, Choice of measure source terms in interface coupling for a model problem in gas dynamics, Math. Comp., 85 (2016), 2305-2339.  doi: 10.1090/mcom/3063.

[28]

A. CorliM. FigielA. Futa and M. D. Rosini, Coupling conditions for isothermal gas flow and applications to valves, Nonlinear Anal. Real World Appl., 40 (2018), 403-427.  doi: 10.1016/j.nonrwa.2017.09.005.

[29]

C. M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl., 38 (1972), 33-41.  doi: 10.1016/0022-247X(72)90114-X.

[30]

C. M. Dafermos, Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method, Arch. Rational Mech. Anal., 52 (1973), 1-9.  doi: 10.1007/BF00249087.

[31]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Volume 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, fourth edition, 2016. doi: 10.1007/978-3-662-49451-6.

[32]

G. Dal MasoP. G. LeFloch and F. Murat, Definition and weak stability of nonconservative products, J. Math. Pures Appl., 74 (1995), 483-548. 

[33]

S. Diehl, Scalar conservation laws with discontinuous flux function.Ⅰ. The viscous profile condition, Comm. Math. Phys., 176 (1996), 23-44.  doi: 10.1007/BF02099361.

[34]

S. Diehl and N.-O. Wallin, Scalar conservation laws with discontinuous flux function. Ⅱ. On the stability of the viscous profiles, Comm. Math. Phys., 176 (1996), 45-71.  doi: 10.1007/BF02099362.

[35]

F. Dubois and P. LeFloch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differential Equations, 71 (1988), 93-122.  doi: 10.1016/0022-0396(88)90040-X.

[36]

T. Galié, Interface Model Coupling in Fluid Dynamics, Application to Two-Phase Flows, Ph.D thesis, Université Pierre et Marie Curie - Paris VI, available from: https://tel.archives-ouvertes.fr/tel-00395593/, 2009.

[37]

M. Garavello and P. Goatin, The Aw-Rascle traffic model with locally constrained flow, J. Math. Anal. Appl., 378 (2011), 634-648.  doi: 10.1016/j.jmaa.2011.01.033.

[38]

M. Garavello and B. Piccoli, Traffic flow on a road network using the Aw-Rascle model, Comm. Partial Differential Equations, 31 (2006), 243-275.  doi: 10.1080/03605300500358053.

[39]

P. Goatin and P. G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 881–902. doi: 10.1016/j.anihpc.2004.02.002.

[40]

E. GodlewskiK.-C. Le Thanh and P.-A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. Ⅱ. The case of systems, M2AN Math. Model. Numer. Anal., 39 (2005), 649-692.  doi: 10.1051/m2an:2005029.

[41]

E. Godlewski and P.-A. Raviart, Hyperbolic Systems of Conservation Laws, Vol. 3/4 of Mathématiques & Applications, Ellipses, Paris, 1991.

[42]

E. Godlewski and P.-A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. Ⅰ. The scalar case, Numer. Math., 97 (2004), 81-130.  doi: 10.1007/s00211-002-0438-5.

[43]

M. Herty, Modeling, simulation and optimization of gas networks with compressors, Netw. Heterog. Media, 2 (2007), 81-97.  doi: 10.3934/nhm.2007.2.81.

[44]

T. Y. Hou and P. G. LeFloch, Why nonconservative schemes converge to wrong solutions. Error analysis, Math. of Comput., 62 (1994), 497-530.  doi: 10.1090/S0025-5718-1994-1201068-0.

[45]

E. Isaacson and B. Temple, Nonlinear resonance in systems of conservation laws, SIAM J. Appl. Math., 52 (1992), 1260-1278.  doi: 10.1137/0152073.

[46]

K. T. Joseph and P. G. LeFloch, Boundary layers in weak solutions to hyperbolic conservation laws, Arch. Rational Mech Anal., 147 (1999), 47-88.  doi: 10.1007/s002050050145.

[47]

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Figure 1.  Illustration of case (ⅱ) of the Corollary 1. The solution $ u $ is continuous at $ \xi = 0 $ (left) vs. discontinuous at $ \xi = 0 $ (right)
Figure 2.  Standing shocks and non trivial inner solution
Figure 3.  Non matching property
Figure 4.  Structure of double-rarefaction solutions
Figure 5.  Structure of double-shock solutions
Figure 6.  Candidate CRD solutions for two convex quadratic fluxes (case $ c>0 $)
Figure 7.  Candidate CRD solutions for two convex quadratic fluxes (case $ c<0 $)
Figure 8.  Numerical approximation of the double-shock solution. CRD solution $ u $ (left) and selection criterion $ h(\cdot;u) $ (right)
Figure 9.  Numerical approximation of the double-rarefaction solution. CRD solution $ u $ (left) and selection criterion $ h(\cdot;u) $ (right)
Figure 10.  Three admissible solutions (from top to bottom). CRD solution $ u $ (left) and selection criterion $ h(\cdot;u) $ (right)
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