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Numerical Approximation of Hyperbolic Systems of Conservation Laws

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Abstract

We now consider general systems of p balance laws

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References

  1. R. Abgrall, How to prevent pressure oscillations in multicomponent flow calculations: a quasi-conservative approach. J. Comput. Phys. 125, 150–160 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  2. R. Abgrall, L. Fezoui, J. Talendier, An extension of Osher’s Riemann solver for chemical and vibrational non-equilibrium gas flows. Int. J. Numer. Methods Fluids 14(8), 935–960 (1992)

    Article  MATH  Google Scholar 

  3. R. Abgrall, S. Karni, Computations of compressible multifluids. J. Comput. Phys. 169, 594–623 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. R. Abgrall, S. Karni, Two-layer shallow water system: a relaxation approach. SIAM J. Sci. Comput. 31, 1603–1627 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  5. R. Abgrall, S. Karni, A comment on the computation of non-conservative products. J. Comput. Phys. 229, 2759–2763 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. R. Abgrall, R. Saurel, Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186, 361–396 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Adimurthi, S. Mishra, G.D.V. Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions. J. Hyperbolic Differ. Equ. 2, 783–837 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  8. N. Aguillon, F. Lagoutière, N. Seguin, Convergence of finite volumes schemes for the coupling between the inviscid Burgers equation and a particle. Math. Comput. 86, 157–196 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. F. Alcrudo, F. Benkhaldoun, Exact solutions to the Riemann problem of the shallow water equations with a bottom step. Comput. Fluids 30, 643–671 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. F. Alouges, B. Merlet, Approximate shock curves for non-conservative hyperbolic systems in one space dimension. J. Hyperbolic Differ. Equ. 1, 769–788 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. D. Amadori, P. Baiti, A. Corli, E. Dal Santo, Global weak solutions for a model of two-phase flow with a single interface. J. Evol. Equ. 15, 699–726 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. D. Amadori, A. Corli, On a model of multiphase flow. SIAM J. Math. Anal. 40, 134–166 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. D. Amadori, L. Gosse, Transient L1 error estimates for well-balanced schemes on non-resonant scalar balance laws. J. Differ. Equ. 255, 469–502 (2013)

    Article  MATH  Google Scholar 

  14. D. Amadori, L. Gosse, G. Guerra, Godunov-type approximation for a general resonant balance law with large data. J. Differ. Equ. 198, 233–274 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  15. L. Ambrosio, B. Alberto, H. Dirk, E. Zuazua, Modelling and Optimisation of Flows on Networks, ed. by B. Piccoli, M. Rascle. C.I.M.E. Foundation Subseries (Springer, Berlin, 2013). Cetraro, Italy 2009

    Google Scholar 

  16. A. Ambroso, C. Chalons, F. Coquel, T. Galié, E. Godlewski, P.-A. Raviart, N. Seguin, The drift-flux asymptotic limit of barotropic two-phase two-pressure models. Commun. Math. Sci. 6, 521–529 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutière, P.-A. Raviart, N. Seguin, The coupling of homogeneous models for two-phase flows. Int. J. Finite 4, 39 (2007)

    MathSciNet  MATH  Google Scholar 

  18. A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutière, P.-A. Raviart, N. Seguin, Coupling of general Lagrangian systems. Math. Comput. 77, 909–941 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. A. Ambroso, C. Chalons, F. Coquel, E. Godlewski, F. Lagoutière, P.-A. Raviart, N. Seguin, Relaxation methods and coupling procedures. Int. J. Numer. Methods Fluids 56, 1123–1129 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. B. Andreianov, The semigroup approach to conservation laws with discontinuous flux, in Hyperbolic Conservation Laws and Related Analysis with Applications, ed. by G.-Q.G. Chen, H. Holden, K.H. Karlsen. Springer Proceedings in Mathematics & Statistics, 2011, pp. 1–22

    Google Scholar 

  21. B. Andreianov, K.H. Karlsen, N.H. Risebro, A theory of L1-dissipative solvers for scalar conservation laws with discontinuous flux. Arch. Ration. Mech. Anal. 201, 27–86 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. N. Andrianov, Performance of numerical methods on the non-unique solution to the Riemann problem for the shallow water equations. Int. J. Numer. Methods Fluids 47, 825–831 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  24. N. Andrianov, G. Warnecke, On the solution to the Riemann problem for the compressible duct flow. SIAM J. Appl. Math. 64, 878–901 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. N. Andrianov, G. Warnecke, The Riemann problem for the Baer-Nunziato two-phase flow model. J. Comput. Phys. 195, 434–464 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. E. Audusse, F. Benkhaldoun, S. Sari, M. Seaid, P. Tassi, A fast finite volume solver for multi-layered shallow water flows with mass exchange. J. Comput. Phys. 272, 23–45 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  27. E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein, B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25, 2050–2065 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  28. E. Audusse, F. Bouchut, M.-O. Bristeau, J. Sainte-Marie, Kinetic entropy inequality and hydrostatic reconstruction scheme for the Saint-Venant system. Math. Comput. 85, 2815–2837 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. E. Audusse, M.-O. Bristeau, M. Pelanti, J. Sainte-Marie, Approximation of the hydrostatic Navier-Stokes system for density stratified flows by a multilayer model: kinetic interpretation and numerical solution. J. Comput. Phys. 230, 3453–3478 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. E. Audusse, M.-O. Bristeau, B. Perthame, J. Sainte-Marie, A multilayer Saint-Venant system with mass exchanges for shallow water flows. Derivation and numerical validation. ESAIM Math. Model. Numer. Anal. 45, 169–200 (2011)

    Article  MATH  Google Scholar 

  31. E. Audusse, B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies. Proc. R. Soc. Edinb. Sect. A 135, 253–265 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  32. F. Bachmann, J. Vovelle, Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients. Commun. Partial Differ. Equ. 31, 371–395 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  33. P. Bagnerini, R.M. Colombo, A. Corli, On the role of source terms in continuum traffic flow models. Math. Comput. Model. 44, 917–930 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  34. J.W. Banks, D.W. Schwendeman, A.K. Kapila, W.D. Henshaw, A high-resolution Godunov method for compressible multi-material flow on overlapping grids. J. Comput. Phys. 223, 262–297 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. F. Béreux, Zero-relaxation limit versus operator splitting for two-phase fluid flow computations. Comput. Methods Appl. Mech. Eng. 133, 93–124 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  36. F. Béreux, L. Sainsaulieu, A Roe-type Riemann solver for hyperbolic systems with relaxation based on time-dependent wave decomposition. Numer. Math. 77, 143–185 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  37. A. Bermúdez, X. López, M.E. Vázquez-Cendón, Numerical solution of non-isothermal non-adiabatic flow of real gases in pipelines. J. Comput. Phys. 323, 126–148 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  38. A. Bermúdez, M.E. Vázquez, Upwind methods for hyperbolic conservation laws with source terms. Comput. Fluids 23, 1049–1071 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  39. A. Bernard-Champmartin, O. Poujade, J. Mathiaud, J.-M. Ghidaglia, Modelling of an homogeneous equilibrium mixture model (HEM). Acta Appl. Math. 129, 1–21 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  40. R. Bernetti, V.A. Titarev, E.F. Toro, Exact solution of the Riemann problem for the shallow water equations with discontinuous bottom geometry. J. Comput. Phys. 227, 3212–3243 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  41. F. Berthelin, J. Vovelle, A Bhatnagar-Gross-Krook approximation to scalar conservation laws with discontinuous flux. Proc. R. Soc. Edinb. Sect. A 140, 953–972 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  42. C. Berthon, M. Bessemoulin-Chatard, H. Mathis, Numerical convergence rate for a diffusive limit of hyperbolic systems: p-system with damping. SMAI J. Comput. Math. 2, 99–119 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  43. C. Berthon, C. Chalons, A fully well-balanced, positive and entropy-satisfying Godunov-type method for the shallow-water equations. Math. Comput. 85, 1281–1307 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  44. C. Berthon, C. Chalons, R. Turpault, Asymptotic-preserving Godunov-type numerical schemes for hyperbolic systems with stiff and nonstiff relaxation terms. Numer. Methods Partial Differ. Equ. 29, 1149–1172 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  45. C. Berthon, P. Charrier, B. Dubroca, An HLLC scheme to solve the M1 model of radiative transfer in two space dimensions. J. Sci. Comput. 31, 347–389 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  46. C. Berthon, F. Coquel, P.G. LeFloch, Why many theories of shock waves are necessary: kinetic relations for non-conservative systems. Proc. R. Soc. Edinburgh Sect. A 142, 1–37 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  47. C. Berthon, F. Foucher, Efficient well-balanced hydrostatic upwind schemes for shallow-water equations. J. Comput. Phys. 231, 4993–5015 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  48. C. Berthon, P.G. LeFloch, R. Turpault, Late-time/stiff-relaxation asymptotic-preserving approximations of hyperbolic equations. Math. Comput. 82, 831–860 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  49. C. Berthon, F. Marche, A positive preserving high order VFRoe scheme for shallow water equations: a class of relaxation schemes. SIAM J. Sci. Comput. 30, 2587–2612 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  50. C. Berthon, F. Marche, R. Turpault, An efficient scheme on wet/dry transitions for shallow water equations with friction. Comput. Fluids 48, 192–201 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  51. C. Berthon, R. Turpault, Asymptotic preserving HLL schemes. Numer. Methods Partial Differ. Equ. 27, 1396–1422 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  52. S. Bianchini, A. Bressan, Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. (2) 161, 223–342 (2005)

    Google Scholar 

  53. S. Boscarino, P.G. LeFloch, G. Russo, High-order asymptotic-preserving methods for fully nonlinear relaxation problems. SIAM J. Sci. Comput. 36, A377–A395 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  54. S. Boscarino, G. Russo, Flux-explicit IMEX Runge-Kutta schemes for hyperbolic to parabolic relaxation problems. SIAM J. Numer. Anal. 51, 163–190 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  55. R. Botchorishvili, B. Perthame, A. Vasseur, Equilibrium schemes for scalar conservation laws with stiff sources. Math. Comput. 72, 131–157 (2003) (electronic)

    Google Scholar 

  56. F. Bouchut, Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Frontiers in Mathematics (Birkhäuser, Basel, 2004)

    Google Scholar 

  57. F. Bouchut, Y. Brenier, J. Cortes, J.-F. Ripoll, A hierarchy of models for two-phase flows. J. Nonlinear Sci. 10, 639–660 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  58. F. Bouchut, E.D. Fernández-Nieto, A. Mangeney, G. Narbona-Reina, A two-phase shallow debris flow model with energy balance. ESAIM Math. Model. Numer. Anal. 49, 101–140 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  59. F. Bouchut, F. James, One-dimensional transport equations with discontinuous coefficients. Nonlinear Anal. 32, 891–933 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  60. F. Bouchut, T. Morales de Luna, An entropy satisfying scheme for two-layer shallow water equations with uncoupled treatment. M2AN Math. Model. Numer. Anal. 42, 683–698 (2008)

    Google Scholar 

  61. F. Bouchut, H. Ounaissa, B. Perthame, Upwinding of the source term at interfaces for Euler equations with high friction. Comput. Math. Appl. 53, 361–375 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  62. F. Bouchut, M. Westdickenberg, Gravity driven shallow water models for arbitrary topography. Commun. Math. Sci. 2, 359–389 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  63. A.-C. Boulanger, C. Cancès, H. Mathis, K. Saleh, and N. Seguin, OSAMOAL: Optimized Simulations by Adapted MOdels using Asymptotic Limits, in CEMRACS’11: Multiscale Coupling of Complex Models in Scientific Computing, vol. 38 of ESAIM Proc., EDP Sci., Les Ulis, 2012, pp. 183–201

    Google Scholar 

  64. C. Bourdarias, S. Gerbi, A finite volume scheme for a model coupling free surface and pressurised flows in pipes. J. Comput. Appl. Math. 209, 109–131 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  65. C. Bourdarias, S. Gerbi, M. Gisclon, A kinetic formulation for a model coupling free surface and pressurised flows in closed pipes. J. Comput. Appl. Math. 218, 522–531 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  66. B. Boutin, C. Chalons, P.-A. Raviart, Existence result for the coupling problem of two scalar conservation laws with Riemann initial data. Math. Models Methods Appl. Sci. 20, 1859–1898 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  67. B. Boutin, F. Coquel, E. Godlewski, Dafermos regularization for interface coupling of conservation laws, in Hyperbolic Problems: Theory, Numerics, Applications (Springer, Berlin, 2008), pp. 567–575

    MATH  Google Scholar 

  68. B. Boutin, F. Coquel, P.G. LeFloch, Coupling techniques for nonlinear hyperbolic equations. III. The well-balanced approximation of thick interfaces. SIAM J. Numer. Anal. 51, 1108–1133 (2013)

    MATH  Google Scholar 

  69. M. Boutounet, L. Chupin, P. Noble, J.P. Vila, Shallow water viscous flows for arbitrary topography. Commun. Math. Sci. 6, 29–55 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  70. A. Bressan, S. Čanić, M. Garavello, M. Herty, B. Piccoli, Flows on networks: recent results and perspectives. EMS Surv. Math. Sci. 1, 47–111 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  71. A. Bressan, D. Serre, M. Williams, K. Zumbrun, Hyperbolic Systems of Balance Laws. Lecture Notes in Mathematics, vol. 1911 (Springer, Berlin, 2007). Fondazione C.I.M.E., Florence, 2007. Lectures given at the C.I.M.E. Summer School held in Cetraro, July 14–21, 2003, Edited and with a preface by Pierangelo Marcati

    Google Scholar 

  72. M.-O. Bristeau, N. Goutal, J. Sainte-Marie, Numerical simulations of a non-hydrostatic shallow water model. Comput. Fluids 47, 51–64 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  73. M.-O. Bristeau, A. Mangeney, J. Sainte-Marie, N. Seguin, An energy-consistent depth-averaged Euler system: derivation and properties. Discrete Contin. Dyn. Syst. Ser. B 20, 961–988 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  74. M.-O. Bristeau, J. Sainte-Marie, Derivation of a non-hydrostatic shallow water model; comparison with Saint-Venant and Boussinesq systems. Discrete Contin. Dyn. Syst. Ser. B 10, 733–759 (2008)

    MathSciNet  MATH  Google Scholar 

  75. S. Bryson, Y. Epshteyn, A. Kurganov, G. Petrova, Well-balanced positivity preserving central-upwind scheme on triangular grids for the Saint-Venant system. ESAIM Math. Model. Numer. Anal. 45, 423–446 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  76. C. Buet, S. Cordier, An asymptotic preserving scheme for hydrodynamics radiative transfer models: numerics for radiative transfer. Numer. Math. 108, 199–221 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  77. C. Buet, B. Després, Asymptotic preserving and positive schemes for radiation hydrodynamics. J. Comput. Phys. 215, 717–740 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  78. T. Buffard, T. Gallouët, J.-M. Hérard, Un schéma simple pour les équations de Saint-Venant, C. R. Acad. Sci. Paris Sér. I Math. 326, 385–390 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  79. R. Bürger, K.H. Karlsen, N.H. Risebro, J.D. Towers, Well-posedness in BVt and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units. Numer. Math. 97, 25–65 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  80. R. Bürger, K.H. Karlsen, J.D. Towers, A conservation law with discontinuous flux modelling traffic flow with abruptly changing road surface conditions, in Hyperbolic Problems: Theory, Numerics and Applications. Proceedings of Symposia in Applied Mathematics. vol. 67 (Amer. Math. Soc., Providence, 2009), pp. 455–464

    Google Scholar 

  81. E. Burman, L. Sainsaulieu, Numerical analysis of two operator splitting methods for an [a] hyperbolic system of conservation laws with stiff relaxation terms. Comput. Methods Appl. Mech. Eng. 128, 291–314 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  82. F. Caetano, Sur certains problèmes de linéarisation et de couplage pour les systèmes hyperboliques non linéaires (French), Ph.D. thesis, UPMC-Paris06 (France), 2006

    Google Scholar 

  83. C. Cancès, F. Coquel, E. Godlewski, H. Mathis, N. Seguin, Error analysis of a dynamic model adaptation procedure for nonlinear hyperbolic equations. Commun. Math. Sci. (2015). 10.4310/CMS.2016.v14.n1.a1

    Google Scholar 

  84. C. Carasso, P.-A. Raviart, D. Serre, eds., Nonlinear Hyperbolic Problems, Proceedings St. Etienne 1986. Lecture Notes in Mathematics, vol. 1270 (Springer, Berlin, 1987)

    Google Scholar 

  85. P. Cargo, A.Y. LeRoux, Un schéma équilibre adapté au modèle d’atmosphère avec termes de gravité. C. R. Acad. Sci. Paris Sér. I Math. 318, 73–76 (1994)

    MathSciNet  MATH  Google Scholar 

  86. C.E. Castro, E.F. Toro, A Riemann solver and upwind methods for a two-phase flow model in non-conservative form. Int. J. Numer. Methods Fluids 50, 275–307 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  87. M.J. Castro Díaz, U.S. Fjordholm, S. Mishra, C. Parés, Entropy conservative and entropy stable schemes for nonconservative hyperbolic systems. SIAM J. Numer. Anal. 51, 1371–1391 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  88. M.J. Castro Díaz, J.M. Gallardo, A. Marquina, Approximate Osher-Solomon schemes for hyperbolic systems. Appl. Math. Comput. 272, 347–368 (2016)

    MathSciNet  MATH  Google Scholar 

  89. M.J. Castro Díaz, J.M. González-Vida, C. Parés, Numerical treatment of wet/dry fronts in shallow flows with a modified Roe scheme. Math. Models Methods Appl. Sci. 16, 897–931 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  90. M.J. Castro Díaz, P.G. LeFloch, M.L. Muñoz-Ruiz, C. Parés, Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes. J. Comput. Phys. 227, 8107–8129 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  91. M.J. Castro Díaz, C. Parés, G. Puppo, G. Russo, Central schemes for nonconservative hyperbolic systems. SIAM J. Sci. Comput. 34, B523–B558 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  92. M.J. Castro Díaz, T. Chacón Rebollo, E.D. Fernández-Nieto, J.M. González Vida, C. Parés, Well-balanced finite volume schemes for 2D non-homogeneous hyperbolic systems. Application to the dam break of Aznalcóllar. Comput. Methods Appl. Mech. Eng. 197, 3932–3950 (2008)

    Article  MATH  Google Scholar 

  93. M.J. Castro Díaz, E.D. Fernández-Nieto, J.M. González-Vida, C. Parés-Madroñal, Numerical treatment of the loss of hyperbolicity of the two-layer shallow-water system. J. Sci. Comput. 48, 16–40 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  94. M.J. Castro Díaz, E.D. Fernández-Nieto, T. Morales de Luna, G. Narbona-Reina, C. Parés, A HLLC scheme for nonconservative hyperbolic problems. Application to turbidity currents with sediment transport. ESAIM Math. Model. Numer. Anal. 47, 1–32 (2013)

    Article  MATH  Google Scholar 

  95. M.J. Castro Díaz, J.A. López-García, C. Parés, High order exactly well-balanced numerical methods for shallow water systems. J. Comput. Phys. 246, 242–264 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  96. A. Chalabi, D. Seghir, Convergence of relaxation schemes for initial boundary value problems for conservation laws. Comput. Math. Appl. 43(8-9), 1079–1093 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  97. N. Chalmers, E. Lorin, Approximation of nonconservative hyperbolic systems based on different shock curve definitions. Can. Appl. Math. Q. 17, 447–485 (2009)

    MathSciNet  MATH  Google Scholar 

  98. N. Chalmers, E. Lorin, On the numerical approximation of one-dimensional nonconservative hyperbolic systems. J. Comput. Sci. 4, 111–124 (2013)

    Article  Google Scholar 

  99. C. Chalons, F. Coquel, E. Godlewski, P.-A. Raviart, N. Seguin, Godunov-type schemes for hyperbolic systems with parameter-dependent source. The case of Euler system with friction. Math. Models Methods Appl. Sci. 20, 2109–2166 (2010)

    Article  MATH  Google Scholar 

  100. C. Chalons, M. Girardin, S. Kokh, Large time step and asymptotic preserving numerical schemes for the gas dynamics equations with source terms. SIAM J. Sci. Comput. 35, A2874–A2902 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  101. C. Chalons, M. Girardin, S. Kokh, Operator-splitting based asymptotic preserving scheme for the gas dynamics equations with stiff source terms, in Proceedings of the 2012 International Conference on Hyperbolic Problems: Theory, Numerics, Applications, no. 8 in AIMS on Applied Mathematics (American Institute of Mathematical Sciences, 2014), pp. 607–614

    Google Scholar 

  102. C. Chalons, P.-A. Raviart, N. Seguin, The interface coupling of the gas dynamics equations. Q. Appl. Math. 66, 659–705 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  103. G.Q. Chen, C.D. Levermore, T.-P. Liu, Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math. 47, 787–830 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  104. A. Chertock, S. Cui, A. Kurganov, C.N. Özcan, E. Tadmor, Well-balanced schemes for the Euler equations with gravitation: conservative formulation using global fluxes. J. Comput. Phys. 358, 36–52 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  105. A. Chinnayya, A.-Y. LeRoux, N. Seguin, A well-balanced numerical scheme for the approximation of the shallow-water equations with topography: the resonance phenomenon. Int. J. Finite 1, 33 (2004)

    MathSciNet  Google Scholar 

  106. A.J. Chorin, M.F. McCracken, T.J.R. Hughes, J.E. Marsden, Product formulas and numerical algorithms. Commun. Pure Appl. Math. 31, 205–256 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  107. S. Clain, D. Rochette, First- and second-order finite volume methods for the one-dimensional nonconservative Euler system. J. Comput. Phys. 228, 8214–8248 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  108. P. Colella, A. Majda, V. Roytburd, Fractional step methods for reacting shock waves, in Reacting Flows: Combustion and Chemical Reactors, Part 2 (Ithaca, N.Y., 1985). Lectures in Appl. Math., vol. 24 (Amer. Math. Soc., Providence, 1986), pp. 459–477

    Google Scholar 

  109. R.M. Colombo, A. Corli, On the operator splitting method: nonlinear balance laws and a generalization of Trotter-Kato formulas, in Hyperbolic problems and regularity questions, Trends Math. (Birkhäuser, Basel, 2007), pp. 91–100

    Google Scholar 

  110. R.M. Colombo, M. Garavello, A well posed Riemann problem for the p-system at a junction. Netw. Heterog. Media 1, 495–511 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  111. R.M. Colombo, G. Guerra, A coupling between a non-linear 1D compressible-incompressible limit and the 1D p-system in the non smooth case. Netw. Heterog. Media 11, 313–330 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  112. R.M. Colombo, M. Herty, Nodal conditions for hyperbolic systems of balance laws, in Hyperbolic Problems: Theory, Numerics, Applications, no. 8 in AIMS on Applied Mathematics (American Institute of Mathematical Sciences, 2014), pp. 147–161. Proceedings of the fourteenth International Conference on Hyperbolic Problems

    Google Scholar 

  113. R.M. Colombo, F. Marcellini, Coupling conditions for the 3 × 3 Euler system. Netw. Heterog. Media 5, 675–690 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  114. R.M. Colombo, F. Marcellini, Smooth and discontinuous junctions in the p-system and in the 3 × 3 Euler system. Riv. Math. Univ. Parma (N.S.) 3, 55–69 (2012)

    Google Scholar 

  115. R.M. Colombo, M. Mercier, M.D. Rosini, Stability and total variation estimates on general scalar balance laws. Commun. Math. Sci. 7, 37–65 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  116. F. Coquel, T. Gallouët, P. Helluy, J.-M. Hérard, O. Hurisse, N. Seguin, Modelling compressible multiphase flows, in Applied mathematics in Savoie—AMIS 2012: Multiphase Flow in Industrial and Environmental Engineering, of ESAIM Proc., EDP Sci., Les Ulis, 2013, pp. 34–50

    Google Scholar 

  117. F. Coquel, E. Godlewski, K. Haddaoui, C. Marmignon, F. Renac, Choice of measure source terms in interface coupling for a model problem in gas dynamics. Math. Comput. 85, 2305–2339 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  118. F. Coquel, J.-M. Hérard, K. Saleh, A splitting method for the isentropic Baer-Nunziato two-phase flow model, in CEMRACS’11: Multiscale Coupling of Complex Models in Scientific Computing, vol. 38 of ESAIM Proc., EDP Sci., Les Ulis, 2012, pp. 241–256

    Google Scholar 

  119. F. Coquel, J.-M. Hérard, K. Saleh, A positive and entropy-satisfying finite volume scheme for the Baer-Nunziato model. J. Comput. Phys. 330, 401–435 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  120. F. Coquel, J.-M. Hérard, K. Saleh, N. Seguin, A robust entropy-satisfying finite volume scheme for the isentropic Baer-Nunziato model. ESAIM Math. Model. Numer. Anal. 48, 165–206 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  121. F. Coquel, S. Jin, J.-G. Liu, L. Wang, Well-posedness and singular limit of a semilinear hyperbolic relaxation system with a two-scale discontinuous relaxation rate. Arch. Ration. Mech. Anal. 214, 1051–1084 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  122. F. Coquel, K. Saleh, N. Seguin, A robust and entropy-satisfying numerical scheme for fluid flows in discontinuous nozzles. Math. Models Methods Appl. Sci. 24, 2043–2083 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  123. 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, 1–9 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  124. G. Dal Maso, P.G. Lefloch, F. Murat, Definition and weak stability of nonconservative products. J. Math. Pures Appl. (9) 74, 483–548 (1995)

    Google Scholar 

  125. A. Decoene, L. Bonaventura, E. Miglio, F. Saleri, Asymptotic derivation of the section-averaged shallow water equations for natural river hydraulics. Math. Models Methods Appl. Sci. 19, 387–417 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  126. P. Degond, G. Dimarco, L. Mieussens, A moving interface method for dynamic kinetic-fluid coupling. J. Comput. Phys. 227, 1176–1208 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  127. P. Degond, S. Jin, A smooth transition model between kinetic and diffusion equations. SIAM J. Numer. Anal. 42, 2671–2687 (2005) (electronic)

    Google Scholar 

  128. M. Deininger, J. Jung, R. Skoda, P. Helluy, C.-D. Munz, Evaluation of interface models for 3D-1D coupling of compressible Euler methods for the application on cavitating flows, in CEMRACS’11: Multiscale Coupling of Complex Models in Scientific Computing, vol. 38 of ESAIM Proc., EDP Sci., Les Ulis, 2012, pp. 298–318

    Google Scholar 

  129. O. Delestre, S. Cordier, F. Darboux, F. James, A limitation of the hydrostatic reconstruction technique for Shallow Water equations. C. R. Math. Acad. Sci. Paris 350, 677–681 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  130. O. Delestre, F. Marche, A numerical scheme for a viscous shallow water model with friction. J. Sci. Comput. 48, 41–51 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  131. V. Desveaux, M. Zenk, C. Berthon, C. Klingenberg, A well-balanced scheme to capture non-explicit steady states in the Euler equations with gravity. Int. J. Numer. Methods Fluids 81, 104–127 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  132. V. Dolejší, T. Gallouët, A numerical study of a particular non-conservative hyperbolic problem. Comput. Fluids 37, 1077–1091 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  133. M. Dumbser, D.S. Balsara, A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems. J. Comput. Phys. 304, 275–319 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  134. M. Dumbser, C. Enaux, E.F. Toro, Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J. Comput. Phys. 227, 3971–4001 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  135. M. Dumbser, E.F. Toro, A simple extension of the Osher Riemann solver to non-conservative hyperbolic systems. J. Sci. Comput. 48, 70–88 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  136. A. Duran, Q. Liang, F. Marche, On the well-balanced numerical discretization of shallow water equations on unstructured meshes. J. Comput. Phys. 235, 565–586 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  137. H. Fan, S. Jin, Z.-h. Teng, Zero reaction limit for hyperbolic conservation laws with source terms. J. Differ. Equ. 168, 270–294 (2000). Special issue in celebration of Jack K. Hale’s 70th birthday, Part 2 (Atlanta, GA/Lisbon, 1998)

    Google Scholar 

  138. E.D. Fernández-Nieto, D. Bresch, J. Monnier, A consistent intermediate wave speed for a well-balanced HLLC solver. C. R. Math. Acad. Sci. Paris 346, 795–800 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  139. E.D. Fernández-Nieto, J. Marin, J. Monnier, Coupling superposed 1D and 2D shallow-water models: source terms in finite volume schemes. Comput. Fluids 39, 1070–1082 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  140. E.D. Fernández-Nieto, G. Narbona-Reina, Extension of WAF type methods to non-homogeneous shallow water equations with pollutant. J. Sci. Comput. 36, 193–217 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  141. F. Filbet, S. Jin, A class of asymptotic-preserving schemes for kinetic equations and related problems with stiff sources. J. Comput. Phys. 229, 7625–7648 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  142. F. Filbet, A. Rambaud, Analysis of an asymptotic preserving scheme for relaxation systems. ESAIM Math. Model. Numer. Anal. 47, 609–633 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  143. P. Finaud-Guyot, C. Delenne, J. Lhomme, V. Guinot, C. Llovel, An approximate-state Riemann solver for the two-dimensional shallow water equations with porosity. Int. J. Numer. Methods Fluids 62, 1299–1331 (2010)

    MathSciNet  MATH  Google Scholar 

  144. U.S. Fjordholm, S. Mishra, E. Tadmor, Well-balanced and energy stable schemes for the shallow water equations with discontinuous topography. J. Comput. Phys. 230, 5587–5609 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  145. T. Flåtten, H. Lund, Relaxation two-phase flow models and the subcharacteristic condition. Math. Models Methods Appl. Sci. 21, 2379–2407 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  146. T. Flåtten, A. Morin, S.T. Munkejord, Wave propagation in multicomponent flow models. SIAM J. Appl. Math. 70, 2861–2882 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  147. A. Forestier, S. Gavrilyuk, Criterion of hyperbolicity for non-conservative quasilinear systems admitting a partially convex conservation law. Math. Methods Appl. Sci. 34, 2148–2158 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  148. F.G. Fuchs, A.D. McMurry, S. Mishra, N.H. Risebro, K. Waagan, High order well-balanced finite volume schemes for simulating wave propagation in stratified magnetic atmospheres. J. Comput. Phys. 229, 4033–4058 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  149. T. Gallouët, Resonance and nonlinearities, in Hyperbolic Problems: Theory, Numerics, Applications (Springer, Berlin, 2008), pp. 113–124

    Book  MATH  Google Scholar 

  150. T. Gallouët, J.-M. Hérard, O. Hurisse, A.-Y. LeRoux, Well-balanced schemes versus fractional step method for hyperbolic systems with source terms. Calcolo 43, 217–251 (2006)

    Article  MathSciNet  Google Scholar 

  151. T. Gallouët, J.-M. Hérard, N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32, 479–513 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  152. T. Gallouët, R. Herbin, J.-C. Latché, On the weak consistency of finite volumes schemes for conservation laws on general meshes. SeMA J. 76, 581–594 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  153. M. Garavello, B. Piccoli, Traffic Flow on Networks. AIMS Series on Applied Mathematics, vol. 1 (American Institute of Mathematical Sciences (AIMS), Springfield, 2006). Conservation laws models

    Google Scholar 

  154. M. Garavello, B. Piccoli, Conservation laws on complex networks. Ann. Inst. H. Poincaré Anal. Non Linéaire 26, 1925–1951 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  155. M. Garavello, B. Piccoli, Coupling of Lighthill-Whitham-Richards and phase transition models. J. Hyperbolic Differ. Equ. 10, 577–636 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  156. J.-F. Gerbeau, B. Perthame, Derivation of viscous Saint-Venant system for laminar shallow water; numerical validation. Discrete Contin. Dyn. Syst. Ser. B 1, 89–102 (2001)

    MathSciNet  MATH  Google Scholar 

  157. P. Goatin, P.G. LeFloch, The Riemann problem for a class of resonant hyperbolic systems of balance laws. Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 881–902 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  158. E. Godlewski, K.-C. Le Thanh, 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, 649–692 (2005)

    Google Scholar 

  159. E. Godlewski, P.-A. Raviart, The numerical interface coupling of nonlinear hyperbolic systems of conservation laws. I. The scalar case. Numer. Math. 97, 81–130 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  160. L. Gosse, A well-balanced scheme using non-conservative products designed for hyperbolic systems of conservation laws with source terms. Math. Models Methods Appl. Sci. 11, 339–365 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  161. L. Gosse, Computing Qualitatively Correct Approximations of Balance Laws. SIMAI Springer Series, vol. 2 (Springer, Milan, 2013). Exponential-fit, well-balanced and asymptotic-preserving.

    Google Scholar 

  162. L. Gosse, A.-Y. LeRoux, Un schéma-équilibre adapté aux lois de conservation scalaires non-homogènes. C. R. Acad. Sci. Paris Sér. I Math. 323, 543–546 (1996)

    MathSciNet  MATH  Google Scholar 

  163. L. Gosse, G. Toscani, An asymptotic-preserving well-balanced scheme for the hyperbolic heat equations. C. R. Math. Acad. Sci. Paris 334, 337–342 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  164. L. Gosse, G. Toscani, Space localization and well-balanced schemes for discrete kinetic models in diffusive regimes. SIAM J. Numer. Anal. 41, 641–658 (2003) (electronic)

    Google Scholar 

  165. N. Goutal, J. Sainte-Marie, A kinetic interpretation of the section-averaged Saint-Venant system for natural river hydraulics. Int. J. Numer. Methods Fluids 67, 914–938 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  166. M. F. Göz, C.-D. Munz, Approximate Riemann solvers for fluid flow with material interfaces, in Numerical Methods for Wave Propagation (Manchester, 1995). Fluid Mech. Appl., vol. 47 (Kluwer Acad. Publ., Dordrecht, 1998), pp. 211–235

    Google Scholar 

  167. J.M. Greenberg, A.Y. LeRoux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33, 1–16 (1996)

    Article  MathSciNet  Google Scholar 

  168. G. Guerra, Well-posedness for a scalar conservation law with singular nonconservative source, J. Differ. Equ. 206, 438–469 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  169. E. Han, M. Hantke, G. Warnecke, Exact Riemann solutions to compressible Euler equations in ducts with discontinuous cross-section. J. Hyperbolic Differ. Equ. 9, 403–449 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  170. E. Han, M. Hantke, G. Warnecke, Criteria for nonuniqueness of Riemann solutions to compressible duct flows. ZAMM Z. Angew. Math. Mech. 93, 465–475 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  171. E. Han, G. Warnecke, Exact Riemann solutions to shallow water equations. Q. Appl. Math. 72, 407–453 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  172. P. Helluy, J.-M. Hérard, H. Mathis, A well-balanced approximate Riemann solver for compressible flows in variable cross-section ducts. J. Comput. Appl. Math. 236, 1976–1992 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  173. P. Helluy, N. Seguin, Relaxation models of phase transition flows. M2AN Math. Model. Numer. Anal. 40, 331–352 (2006)

    Google Scholar 

  174. C. Helzel, R.J. LeVeque, G. Warnecke, A modified fractional step method for the accurate approximation of detonation waves. SIAM J. Sci. Comput. 22, 1489–1510 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  175. J.-M. Hérard, Un modèle hyperbolique diphasique bi-fluide en milieu poreux. C. R. Mecanique. Acad. Sci. Paris 336, 650–655 (2008)

    Article  MATH  Google Scholar 

  176. J.-M. Hérard, O. Hurisse, Coupling two and one-dimensional unsteady euler equations through a thin interface. Comput. Fluids 5, 651–666 (2007)

    Article  MATH  Google Scholar 

  177. G. Hernández-Dueñas, S. Karni, Shallow water flows in channels. J. Sci. Comput. 48, 190–208 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  178. M. Herty, S. Moutari, M. Rascle, Optimization criteria for modelling intersections of vehicular traffic flow. Netw. Heterog. Media 1, 275–294 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  179. M. Herty, M. Rascle, Coupling conditions for a class of second-order models for traffic flow. SIAM J. Math. Anal. 38, 595–616 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  180. J.A.F. Hittinger, P.L. Roe, Asymptotic analysis of the Riemann problem for constant coefficient hyperbolic systems with relaxation. ZAMM Z. Angew. Math. Mech. 84, 452–471 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  181. H. Holden, K.H. Karlsen, K.-A. Lie, N.H. Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions. EMS Series of Lectures in Mathematics, (European Mathematical Society (EMS), Zürich, 2010). Analysis and MATLAB programs

    Google Scholar 

  182. J. Hong, B. Temple, The generic solution of the Riemann problem in a neighborhood of a point of resonance for systems of nonlinear balance laws. Methods Appl. Anal. 10, 279–294 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  183. L. Hsiao, T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping. Commun. Math. Phys. 143, 599–605 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  184. L. Hsiao, D. Serre, Global existence of solutions for the system of compressible adiabatic flow through porous media. SIAM J. Math. Anal. 27, 70–77 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  185. F. Huang, P. Marcati, R. Pan, Convergence to the Barenblatt solution for the compressible Euler equations with damping and vacuum. Arch. Ration. Mech. Anal. 176, 1–24 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  186. L.P. Huang, T.-P. Liu, A conservative, piecewise-steady difference scheme for transonic nozzle flow, Comput. Math. Appl. A 12, 377–388 (1986). Hyperbolic partial differential equations, III

    Google Scholar 

  187. E.L. Isaacson, J.B. Temple, Nonlinear resonance in systems of conservation laws. SIAM J. Appl. Math. 52, 1260–1278 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  188. E.L. Isaacson, J.B. Temple, Convergence of the 2 × 2 Godunov method for a general resonant nonlinear balance law. SIAM J. Appl. Math. 55, 625–640 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  189. S. Jin, Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21, 441–454 (1999) (electronic)

    Google Scholar 

  190. S. Jin, A steady-state capturing method for hyperbolic systems with geometrical source terms. M2AN Math. Model. Numer. Anal. 35, 631–645 (2001)

    Google Scholar 

  191. S. Jin, Asymptotic preserving (AP) schemes for multiscale kinetic and hyperbolic equations: a review. Riv. Math. Univ. Parma (N.S.) 3, 177–216 (2012)

    Google Scholar 

  192. S. Jin, Y.J. Kim, On the computation of roll waves. M2AN Math. Model. Numer. Anal. 35, 463–480 (2001)

    Google Scholar 

  193. S. Jin, C.D. Levermore, Numerical schemes for hyperbolic conservation laws with stiff relaxation terms. J. Comput. Phys. 126, 449–467 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  194. S. Jin, J.-G. Liu, L. Wang, A domain decomposition method for semilinear hyperbolic systems with two-scale relaxations. Math. Comput. 82, 749–779 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  195. S. Jin, L. Pareschi, G. Toscani, Uniformly accurate diffusive relaxation schemes for multiscale transport equations. SIAM J. Numer. Anal. 38, 913–936 (2000) (electronic)

    Google Scholar 

  196. S. Jin, X. Wen, Two interface-type numerical methods for computing hyperbolic systems with geometrical source terms having concentrations. SIAM J. Sci. Comput. 26, 2079–2101 (2005) (electronic)

    Google Scholar 

  197. S. Jin, Z. P. Xin The relaxation schemes for systems of conservation laws in arbitrary space dimensions. Commun. Pure Appl. Math. 48, 235–276 (1995)

    Google Scholar 

  198. R. Käppeli, S. Mishra, Well-balanced schemes for the Euler equations with gravitation. J. Comput. Phys. 259, 199–219 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  199. K.H. Karlsen, M. Rascle, E. Tadmor, On the existence and compactness of a two-dimensional resonant system of conservation laws. Commun. Math. Sci. 5, 253–265 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  200. K.H. Karlsen, N.H. Risebro, J.D. Towers, Front tracking for scalar balance equations. J. Hyperbolic Differ. Equ. 1, 115–148 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  201. S. Karni, Multicomponent flow calculations by a consistent primitive algorithm. J. Comput. Phys. 112, 31–43 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  202. T. Kato, Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups, in Topics in Functional Analysis (Essays Dedicated to M. G. Kreı̆n on the Occasion of his 70th Birthday). Adv. in Math. Suppl. Stud., vol. 3 (Academic Press, New York-London, 1978), pp. 185–195

    Google Scholar 

  203. A. Klar, Convergence of alternating domain decomposition schemes for kinetic and aerodynamic equations. Math. Methods Appl. Sci. 18, 649–670 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  204. A. Klar, R. Wegener, A hierarchy of models for multilane vehicular traffic. I. Modeling. SIAM J. Appl. Math. 59, 983–1001 (1999) (electronic)

    Google Scholar 

  205. D. Kröner, P.G. LeFloch, M.-D. Thanh, The minimum entropy principle for compressible fluid flows in a nozzle with discontinuous cross-section. M2AN Math. Model. Numer. Anal. 42, 425–442 (2008)

    Google Scholar 

  206. D. Kröner, M.D. Thanh, Numerical solutions to compressible flows in a nozzle with variable cross-section. SIAM J. Numer. Anal. 43, 796–824 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  207. A. Kurganov, G. Petrova, Central-upwind schemes for two-layer shallow water equations. SIAM J. Sci. Comput. 31, 1742–1773 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  208. Y.-S. Kwon, A. Vasseur, Strong traces for solutions to scalar conservation laws with general flux. Arch. Ration. Mech. Anal. 185, 495–513 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  209. M.-H. Lallemand, A. Chinnayya, O. Le Metayer, Pressure relaxation procedures for multiphase compressible flows. Int. J. Numer. Methods Fluids 49, 1–56 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  210. J.O. Langseth, A. Tveito, R. Winther, On the convergence of operator splitting applied to conservation laws with source terms. SIAM J. Numer. Anal. 33, 843–863 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  211. D. Lannes, F. Marche, A new class of fully nonlinear and weakly dispersive Green-Naghdi models for efficient 2D simulations. J. Comput. Phys. 282, 238–268 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  212. C. Lattanzio, B. Piccoli, Coupling of microscopic and macroscopic traffic models at boundaries. Math. Models Methods Appl. Sci. 20, 2349–2370 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  213. C. Lattanzio, A.E. Tzavaras, Relative entropy in diffusive relaxation. SIAM J. Math. Anal. 45, 1563–1584 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  214. P. Le Tallec, F. Mallinger, Coupling Boltzmann and Navier-Stokes equations by half fluxes. J. Comput. Phys. 136, 51–67 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  215. M. Lécureux-Mercier, Improved stability estimates for general scalar conservation laws. J. Hyperbolic Differ. Equ. 8, 727–757 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  216. P.G. LeFloch, M.D. Thanh, The Riemann problem for fluid flows in a nozzle with discontinuous cross-section. Commun. Math. Sci. 1, 763–797 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  217. P.G. LeFloch, M.D. Thanh, The Riemann problem for the shallow water equations with discontinuous topography. Commun. Math. Sci. 5, 865–885 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  218. P.G. Lefloch, A.E. Tzavaras, Representation of weak limits and definition of nonconservative products. SIAM J. Math. Anal. 30, 1309–1342 (1999) (electronic)

    Google Scholar 

  219. A.Y. LeRoux, Riemann solvers for some hyperbolic problems with a source term, in Actes du 30ème Congrès d’Analyse Numérique: CANum ’98 (Arles, 1998), vol. 6 of ESAIM Proc., Soc. Math. Appl. Indust., Paris, 1999, pp. 75–90 (electronic)

    Google Scholar 

  220. A.-Y. LeRoux, L. Vignon, Sur la modélisation de la stabilité d’une colonne atmosphérique avec gravité dépendant de l’altitude. C. R. Math. Acad. Sci. Paris 346, 239–242 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  221. R.J. LeVeque, Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics (Cambridge University Press, Cambridge, 2002)

    Google Scholar 

  222. R.J. LeVeque, M. Pelanti, A class of approximate Riemann solvers and their relation to relaxation schemes. J. Comput. Phys. 172, 572–591 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  223. R.J. LeVeque, H.C. Yee, A study of numerical methods for hyperbolic conservation laws with stiff source terms. J. Comput. Phys. 86, 187–210 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  224. J. Lhomme, V. Guinot, A general approximate-state Riemann solver for hyperbolic systems of conservation laws with source terms. Int. J. Numer. Methods Fluids 53, 1509–1540 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  225. C. Lian, G. Xia, C.L. Merkle, Impact of source terms on reliability of CFD algorithms. Comput. Fluids 39, 1909–1922 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  226. L.W. Lin, B. Temple, J.H. Wang, Suppression of oscillations in Godunov’s method for a resonant non-strictly hyperbolic system. SIAM J. Numer. Anal. 32, 841–864 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  227. X.-B. Lin, S. Schecter, Stability of self-similar solutions of the Dafermos regularization of a system of conservation laws. SIAM J. Math. Anal. 35, 884–921 (2003) (electronic)

    Google Scholar 

  228. P.L. Lions, G. Toscani, Diffusive limit for finite velocity Boltzmann kinetic models. Rev. Mat. Iberoam. 13, 473–513 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  229. T.P. Liu, Nonlinear resonance for quasilinear hyperbolic equation. J. Math. Phys. 28, 2593–2602 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  230. H. Lund, A hierarchy of relaxation models for two-phase flow. SIAM J. Appl. Math. 72, 1713–1741 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  231. P. Marcati, A. Milani, The one-dimensional Darcy’s law as the limit of a compressible Euler flow. J. Differ. Equ. 84, 129–147 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  232. F. Marche, P. Bonneton, P. Fabrie, N. Seguin, Evaluation of well-balanced bore-capturing schemes for 2D wetting and drying processes. Int. J. Numer. Methods Fluids 53, 867–894 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  233. H. Mathis, C. Cancès, E. Godlewski, N. Seguin, Dynamic model adaptation for multiscale simulation of hyperbolic systems with relaxation. J. Sci. Comput. 63, 820–861

    Google Scholar 

  234. R. Menina, R. Saurel, M. Zereg, L. Houas, Modelling gas dynamics in 1d ducts with abrupt area change. Shock Waves 21, 451–466 (2011)

    Article  Google Scholar 

  235. G. Montecinos, C.E. Castro, M. Dumbser, E.F. Toro, Comparison of solvers for the generalized Riemann problem for hyperbolic systems with source terms. J. Comput. Phys. 231, 6472–6494 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  236. T. Morales de Luna, M.J. Castro Díaz, C. Parés, Reliability of first order numerical schemes for solving shallow water system over abrupt topography. Appl. Math. Comput. 219, 9012–9032 (2013)

    MathSciNet  MATH  Google Scholar 

  237. S.T. Munkejord, A numerical study of two-fluid models with pressure and velocity relaxation. Adv. Appl. Math. Mech. 2, 131–159 (2010)

    Article  MathSciNet  Google Scholar 

  238. J. Murillo, P. García-Navarro, Weak solutions for partial differential equations with source terms: application to the shallow water equations. J. Comput. Phys. 229, 4327–4368 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  239. J. Murillo, P. García-Navarro, Augmented versions of the HLL and HLLC Riemann solvers including source terms in one and two dimensions for shallow flow applications. J. Comput. Phys. 231, 6861–6906 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  240. G. Naldi, L. Pareschi, Numerical schemes for hyperbolic systems of conservation laws with stiff diffusive relaxation. SIAM J. Numer. Anal. 37, 1246–1270 (2000) (electronic)

    Google Scholar 

  241. G. Narbona-Reina, J.D.D. Zabsonré, E.D. Fernández-Nieto, D. Bresch, Derivation of a bilayer model for shallow water equations with viscosity. Numerical validation. CMES Comput. Model. Eng. Sci. 43, 27–71 (2009)

    MathSciNet  MATH  Google Scholar 

  242. J. Neusser, V. Schleper, Numerical schemes for the coupling of compressible and incompressible fluids in several space dimensions. Appl. Math. Comput. 304, 65–82 (2017)

    MathSciNet  MATH  Google Scholar 

  243. P. Noble, Roll-waves in general hyperbolic systems with source terms. SIAM J. Appl. Math. 67, 1202–1212 (2007) (electronic)

    Google Scholar 

  244. S. Noelle, N. Pankratz, G. Puppo, J.R. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213, 474–499 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  245. S. Noelle, Y. Xing, C.-W. Shu, High-order well-balanced finite volume WENO schemes for shallow water equation with moving water. J. Comput. Phys. 226, 29–58 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  246. S. Noelle, Y. Xing, C.-W. Shu, High-order well-balanced schemes, in Numerical Methods for Balance Laws. Quad. Mat., vol. 24. Dept. Math., Seconda Univ. Napoli, Caserta (2009), pp. 1–66

    Google Scholar 

  247. A. Noussair, Riemann problem with nonlinear resonance effects and well-balanced Godunov scheme for shallow fluid flow past an obstacle. SIAM J. Numer. Anal. 39, 52–72 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  248. E.Y. Panov, On existence and uniqueness of entropy solutions to the Cauchy problem for a conservation law with discontinuous flux. J. Hyperbolic Differ. Equ. 6, 525–548 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  249. C. Parés, M. Castro, On the well-balance property of Roe’s method for nonconservative hyperbolic systems. Applications to shallow-water systems. M2AN Math. Model. Numer. Anal. 38, 821–852 (2004)

    Google Scholar 

  250. M. Parisot, J.-P. Vila, Numerical scheme for multilayer shallow-water model in the low-Froude number regime. C. R. Math. Acad. Sci. Paris 352, 953–957 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  251. M. Pelanti, F. Bouchut, A. Mangeney, A Roe-type scheme for two-phase shallow granular flows over variable topography. M2AN Math. Model. Numer. Anal. 42, 851–885 (2008)

    Google Scholar 

  252. M. Pelanti, F. Bouchut, A. Mangeney, A Riemann solver for single-phase and two-phase shallow flow models based on relaxation. Relations with Roe and VFRoe solvers. J. Comput. Phys. 230, 515–550 (2011)

    MATH  Google Scholar 

  253. R.B. Pember, Numerical methods for hyperbolic conservation laws with stiff relaxation. I. Spurious solutions. SIAM J. Appl. Math. 53, 1293–1330 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  254. R.B. Pember, Numerical methods for hyperbolic conservation laws with stiff relaxation. II. Higher-order Godunov methods. SIAM J. Sci. Comput. 14, 824–859 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  255. B. Perthame, C. Simeoni, A kinetic scheme for the Saint-Venant system with a source term. Calcolo 38, 201–231 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  256. E.B. Pitman, L. Le, A two-fluid model for avalanche and debris flows. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 363, 1573–1601 (2005)

    MathSciNet  MATH  Google Scholar 

  257. G. Puppo, G. Russo (eds.), Numerical Methods for Balance Laws. Quaderni di Matematica [Mathematics Series], vol. 24. Department of Mathematics, Seconda Università di Napoli, Caserta, 2009

    Google Scholar 

  258. P.-A. Raviart, L. Sainsaulieu, A nonconservative hyperbolic system modeling spray dynamics. I. Solution of the Riemann problem. Math. Models Methods Appl. Sci. 5, 297–333 (1995)

    Article  MATH  Google Scholar 

  259. G.A. Reigstad, T. Flåtten, N. Erland Haugen, T. Ytrehus, Coupling constants and the generalized Riemann problem for isothermal junction flow. J. Hyperbolic Differ. Equ. 12, 37–59 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  260. G.L. Richard, S.L. Gavrilyuk, A new model of roll waves: comparison with Brock’s experiments. J. Fluid Mech. 698, 374–405 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  261. D. Rochette, S. Clain, W. Bussière, Unsteady compressible flow in ducts with varying cross-section: comparison between the nonconservative Euler system and the axisymmetric flow model. Comput. Fluids 53, 53–78 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  262. L. Sainsaulieu, Traveling-wave solutions of convection-diffusion systems in nonconservation form. SIAM J. Math. Anal. 27, 1286–1310 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  263. J. Sainte-Marie, Vertically averaged models for the free surface non-hydrostatic Euler system: derivation and kinetic interpretation. Math. Models Methods Appl. Sci. 21, 459–490 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  264. R. Saurel, R. Abgrall, A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150, 425–467 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  265. R. Saurel, R. Abgrall, A simple method for compressible multifluid flows. SIAM J. Sci. Comput. 21, 1115–1145 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  266. H.J. Schroll, R. Winther, Finite-difference schemes for scalar conservation laws with source terms. IMA J. Numer. Anal. 16, 201–215 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  267. N. Seguin, J. Vovelle, Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients. Math. Models Methods Appl. Sci. 13, 221–257 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  268. D. Serre, A.F. Vasseur, About the relative entropy method for hyperbolic systems of conservation laws, in A Panorama of Mathematics: Pure and Applied. Contemp. Math., vol. 658 (Amer. Math. Soc., Providence, 2016), pp. 237–248

    Google Scholar 

  269. D. Serre, L. Xiao, Asymptotic behavior of large weak entropy solutions of the damped P-system. J. Partial Differ. Equ. 10, 355–368 (1997)

    MathSciNet  MATH  Google Scholar 

  270. C. Simeoni, Remarks on the consistency of upwind source at interface schemes on nonuniform grids. J. Sci. Comput. 48, 333–338 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  271. H.B. Stewart, B. Wendroff, Two-phase flow: models and methods. J. Comput. Phys. 56, 363–409 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  272. T. Tang, Convergence analysis for operator-splitting methods applied to conservation laws with stiff source terms. SIAM J. Numer. Anal. 35, 1939–1968 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  273. T. Tang, Z.H. Teng, Time-splitting methods for nonhomogeneous conservation laws, in Mathematics of Computation 1943–1993: A Half-century of Computational Mathematics (Vancouver, BC, 1993). Proceedings of the Symposium Application Mathematical, vol. 48 (American Mathematical Society, Providence, 1994), pp. 389–393

    Google Scholar 

  274. T. Tang, Z.H. Teng, Error bounds for fractional step methods for conservation laws with source terms. SIAM J. Numer. Anal. 32, 110–127 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  275. B. Temple, Global solution of the Cauchy problem for a class of 2 × 2 nonstrictly hyperbolic conservation laws. Adv. Appl. Math. 3, 335–375 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  276. Z.H. Teng, First-order L1-convergence for relaxation approximations to conservation laws. Commun. Pure Appl. Math. 51, 857–895 (1998)

    Article  Google Scholar 

  277. M.D. Thanh, Numerical treatment in resonant regime for shallow water equations with discontinuous topography. Commun. Nonlinear Sci. Numer. Simul. 18, 417–433 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  278. M.D. Thanh, D.H. Cuong, Existence of solutions to the Riemann problem for a model of two-phase flows. Electron. J. Differ. Equ. 32, 18 (2015)

    MathSciNet  MATH  Google Scholar 

  279. M.D. Thanh, D. Kröner, Numerical treatment of nonconservative terms in resonant regime for fluid flows in a nozzle with variable cross-section. Comput. Fluids 66, 130–139 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  280. E.F. Toro, Riemann problems and the WAF method for solving the two-dimensional shallow water equations. Philos. Trans. R. Soc. London Ser. A 338, 43–68 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  281. A.E. Tzavaras, Relative entropy in hyperbolic relaxation, Commun. Math. Sci. 3, 119–132 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  282. A.F. Vasseur, Well-posedness of scalar conservation laws with singular sources. Methods Appl. Anal. 9, 291–312 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  283. A.F. Vasseur, A rigorous derivation of the coupling of a kinetic equation and Burgers’ equation. Arch. Ration. Mech. Anal. 206, 1–30 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  284. M.E. Vázquez-Cendón, Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comput. Phys. 148, 497–526 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  285. G. Wu, Z. Tan, J. Huang, Global existence and large time behavior for the system of compressible adiabatic flow through porous media in \(\mathbb {R}^3\). J. Differ. Equ. 255, 865–880 (2013)

    Google Scholar 

  286. Y. Xing, C.-W. Shu, High order finite difference WENO schemes with the exact conservation property for the shallow water equations. J. Comput. Phys. 208, 206–227 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  287. Y. Xing, C.-W. Shu, High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields. J. Sci. Comput. 54, 645–662 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  288. Y. Xing, C.-W. Shu, A survey of high order schemes for the shallow water equations. J. Math. Study 47, 221–249 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  289. Y. Xing, C.-W. Shu, S. Noelle, On the advantage of well-balanced schemes for moving-water equilibria of the shallow water equations. J. Sci. Comput. 48, 339–349 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  290. A. Zein, M. Hantke, G. Warnecke, Modeling phase transition for compressible two-phase flows applied to metastable liquids. J. Comput. Phys. 229, 2964–2998 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Godlewski, E., Raviart, PA. (2021). Source Terms. In: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Applied Mathematical Sciences, vol 118. Springer, New York, NY. https://doi.org/10.1007/978-1-0716-1344-3_7

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