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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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[22]

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[23]

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[26]

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[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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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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