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A random sampling method for a family of Temple-class systems of conservation laws

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Abstract

The Aw–Rascle–Zhang traffic model, a model of sedimentation, and other applications lead to nonlinear \(2 \times 2\) systems of conservation laws that are governed by a single scalar system velocity. Such systems are of the Temple class since rarefaction wave curves and Hugoniot curves coincide. Moreover, one characteristic field is genuinely nonlinear almost everywhere, and the other is linearly degenerate. Two well-known problems associated with these systems are handled via a random sampling approach. Firstly, Godunov’s and related methods produce spurious oscillations near contact discontinuities since the numerical solution invariably leaves the invariant region of the exact solution. It is shown that alternating between averaging (Av) and remap steps similar to the approach by Chalons and Goatin (Commun Math Sci 5:533–551, 2007) generates numerical solutions that do satisfy an invariant region property. If the remap step is made by random sampling (RS), then combined techniques due to Glimm (Commun Pure Appl Math 18:697–715, 1965), LeVeque and Temple (Trans Am Math Soc 288:115–123, 1985) prove that the resulting Av–RS scheme converges to a weak solution. Numerical examples illustrate that the new scheme is superior to Godunov’s method in accuracy and resolution. Secondly, the vacuum state, which may form even from positive initial data, causes potential problems of non-uniqueness and instability. This is resolved by introducing an alternative Riemann solution concept.

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References

  1. Aw, A., Rascle, M.: Resurrection of “second order” models of traffic flow. SIAM J. Appl. Math. 60(3), 916–938 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bachmann, M., Helluy, P., Jung, J., Mathis, H., Müller, S.: Random sampling remap for compressible two-phase flows. Comput. Fluids 86, 275–283 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Banks, J.W.: A note on the convergence of Godunov type methods for shock reflection problems. Comput. Math. Appl. 66(1), 19–23 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Betancourt, F., Bürger, R., Diehl, S., Farås, S.: Modeling and controlling clarifier-thickeners fed by suspensions with time-dependent properties. Miner. Eng. 62, 91–101 (2014)

    Article  Google Scholar 

  5. Bressan, A., Jenssen, H.K., Baiti, P.: An instability of the Godunov scheme. Commun. Pure Appl. Math. 59(11), 1604–1638 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bürger, R., Chalons, C., Villada, L.M.: Antidiffusive and random-sampling Lagrangian-remap schemes for the multiclass Lighthill–Whitham–Richards traffic model. SIAM J. Sci. Comput. 35(6), B1341–B1368 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chalons, C.: Numerical approximation of a macroscopic model of pedestrian flows. SIAM J. Sci. Comput. 29(2), 539–555 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chalons, C., Coquel, F.: Computing material fronts with a Lagrange-Projection approach. In: Series in Contemporary Applied Mathematics CAM, vol. 1, pp. 346–356. World Scintific Publishing, Singapore (2012). (Proceedings of international Conference HYP2010)

  9. Chalons, C., Goatin, P.: Transport-equilibrium schemes for computing contact discontinuities in traffic flow modeling. Commun. Math. Sci. 5(3), 533–551 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chalons, C., Goatin, P.: Godunov scheme and sampling technique for computing phase transitions in traffic flow modeling. Interfaces Free Bound. 10(2), 197–221 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chalons, C., Goatin, P., Seguin, N.: General constrained conservation laws. Application to pedestrian flow modeling. Netw. Heterog. Media 8(2), 433–463 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Colella, P.: Glimm’s method for gas dynamics. SIAM J. Sci. Stat. Comput. 3(1), 76–110 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. Colombo, R.M.: A \(2 \times 2\) hyperbolic traffic flow model. Math. Comput. Model. 35(5–6), 683–688 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fan, S., Herty, M., Seibold, B.: Comparative model accuracy of a data-fitted generalized Aw–Rascle–Zhang model. Netw. Heterog. Media 9(2), 239–268 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Garavello, M., Goatin, P.: The Aw-Rascle traffic model with locally constrained flow. J. Math. Anal. Appl. 378(2), 634–648 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Glimm, J.: Solutions in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 18(4), 697–715 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  17. Goatin, P.: The Aw-Rascle vehicular traffic flow model with phase transitions. Math. Comput. Model. 44(3–4), 287–303 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  18. Godvik, M., Hanche-Olsen, H.: Existence of solutions for the Aw-Rascle traffic flow model with vacuum. J. Hyperbolic Differ. Equ. 05(01), 45–63 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Harten, A., Lax, P.D., van Leer, B.: On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Rev. 25(1), 35–61 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  20. Holden, H., Risebro, N.H.: Front Tracking for Hyperbolic Conservation Laws. Springer, Berlin Heidelberg (2011)

    Book  MATH  Google Scholar 

  21. Isaacson, E.L., Temple, J.B.: Analysis of a singular hyperbolic system of conservation laws. J. Differ. Equ. 65(2), 250–268 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  22. Keyfitz, B.L., Kranzer, H.C.: A system of non-strictly hyperbolic conservation laws arising in elasticity theory. Arch. Ration. Mech. Anal. 72(3), 219–241 (1980)

    Article  MATH  Google Scholar 

  23. Klingenberg, C., Risebro, N.H.: Stability of a resonant system of conservation laws modeling polymer flow with gravitation. J. Differ. Equ. 170, 344–380 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kokh, S., Lagoutière, F.: An anti-diffusive numerical scheme for the simulation of interfaces between compressible fluids by means of a five-equation model. J. Comput. Phys. 229(8), 2773–2809 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kurganov, A., Lin, C.T.: On the reduction of numerical dissipation in central-upwind schemes. Commun. Comput. Phys. 2(1), 141–163 (2007)

    MathSciNet  MATH  Google Scholar 

  26. Kurganov, A., Tadmor, E.: New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. J. Comput. Phys. 160(1), 241–282 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  27. Kynch, G.J.: A theory of sedimentation. Trans. Faraday Soc. 48, 166–176 (1952)

    Article  Google Scholar 

  28. Lebacque, J.-P., Mammar, S., Haj-Salem, H.: The Aw-Rascle and Zhang’s model: vacuum problems, existence and regularity of the solutions of the Riemann problem. Transp. Res. Part B Methodol. 41(7), 710–721 (2007)

    Article  Google Scholar 

  29. LeVeque, R.J.: Numerical Methods for Conservation Laws. Birkhäuser Verlag, Basel (1992)

    Book  MATH  Google Scholar 

  30. LeVeque, R.J., Temple, B.: Stability of Godunov’s method for a class of \(2 \times 2\) systems of conservation laws. Trans. Am. Math. Soc. 288(1), 115–123 (1985)

    MathSciNet  MATH  Google Scholar 

  31. Li, T.: Global solutions of nonconcave hyperbolic conservation laws with relaxation arising from traffic flow. J. Differ. Equ. 190(1), 131–149 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  32. Lin, L., Temple, J.B., Wang, J.: A comparison of convergence rates for Godunov’s method and Glimm’s method in resonant nonlinear systems of conservation laws. SIAM J. Numer. Anal. 32(3), 824–840 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. Liu, T.-P.: The entropy condition and the admissibility of shocks. J. Math. Anal. Appl. 53(1), 78–88 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  34. Longwei, L., Temple, B., Jinghua, W.: Suppression of oscillations in Godunov’s method for a resonant non-strictly hyperbolic system. SIAM J. Numer. Anal. 32(3), 841–864 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lu, Y.-G., Gu, F.: Existence of global bounded weak solutions to a Keyfitz–Kranzer system. Commun. Math. Sci. 10(4), 1133–1142 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  36. Moutari, S., Rascle, M.: A hybrid Lagrangian model based on the Aw-Rascle traffic flow model. SIAM J. Appl. Math. 68(2), 413–436 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Oleinik, O.A.: Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. Uspekhi Mat. Nauk 14, 165–170 (1959)

    MathSciNet  Google Scholar 

  38. Oleinik, O.A.: Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. Am. Math. Soc. Transl. Ser. 2(33), 285–290 (1964)

    MATH  Google Scholar 

  39. Qiao, D.L., Zhang, P., Wong, S.C., Choi, K.: Discontinuous Galerkin finite element scheme for a conserved higher-order traffic flow model by exploring Riemann solvers. Appl. Math. Comput. 244, 567–576 (2014)

    MathSciNet  MATH  Google Scholar 

  40. Richardson, J.F., Zaki, W.N.: Sedimentation and fluidization: part I. Trans. Inst. Chem. Eng. 32, 35–53 (1954)

    Google Scholar 

  41. Smoller, J.: Shock Waves and Reaction-Diffusion Equations. Springer-Verlag, Berlin (1983)

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  43. Temple, B.: Systems of conservation laws with invariant submanifolds. Trans. Am. Math. Soc. 280(2), 781–781 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  44. Wagner, D.H.: Equivalence of the Euler and Lagrangian equations of gas dynamics for weak solutions. J. Differ. Equ. 68(1), 118–136 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  45. Zhang, H.M.: A non-equilibrium traffic model devoid of gas-like behavior. Transp. Res. Part B Methodol. 36(3), 275–290 (2002)

    Article  Google Scholar 

Download references

Acknowledgements

FB and RB acknowledge support by BASAL project CMM, Universidad de Chile and Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción; Centro CRHIAM Proyecto Conicyt Fondap 15130015; and Fondef project ID15I10291. In addition, FB is supported by Fondecyt project 1130154, and RB is supported by Fondecyt project 1170473 and Conicyt project Anillo ACT1118 (ANANUM).

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Authors and Affiliations

  1. CI2MA and Departamento de Ingeniería Metalúrgica, Facultad de Ingeniería, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Fernando Betancourt

  2. CI2MA and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Raimund Bürger

  3. Laboratoire de Mathématiques de Versailles, UMR 8100, Université de Versailles Saint-Quentin-en-Yvelines, UFR des Sciences, bâtiment Fermat, 45 avenue des Etats-Unis, 78035, Versailles Cedex, France

    Christophe Chalons

  4. Centre for Mathematical Sciences, Lund University, P.O. Box 118, 221 00, Lund, Sweden

    Stefan Diehl & Sebastian Farås

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  1. Fernando Betancourt
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  2. Raimund Bürger
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  3. Christophe Chalons
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  4. Stefan Diehl
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  5. Sebastian Farås
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Correspondence to Sebastian Farås.

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Betancourt, F., Bürger, R., Chalons, C. et al. A random sampling method for a family of Temple-class systems of conservation laws. Numer. Math. 138, 37–73 (2018). https://doi.org/10.1007/s00211-017-0900-z

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  • DOI: https://doi.org/10.1007/s00211-017-0900-z

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