Skip to main content
SpringerLink
Log in

Convergence of the Finite Volume method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation

  • Published:
Stochastics and Partial Differential Equations: Analysis and Computations Aims and scope Submit manuscript

Abstract

Under a standard CFL condition, we prove the convergence of the explicit-in-time Finite Volume method with monotone fluxes for the approximation of scalar first-order conservation laws with multiplicative, compactly supported noise. In Dotti and Vovelle (Arch Ration Mech Anal 230(2):539–591, 2018), a framework for the analysis of the convergence of approximations to stochastic scalar first-order conservation laws has been developed, on the basis of a kinetic formulation. Here, we give a kinetic formulation of the numerical method, analyse its properties, and explain how to cast the problem of convergence of the numerical scheme into the framework of Dotti and Vovelle (2018). This uses standard estimates (like the so-called “weak BV estimate”, for which we give a proof using specifically the kinetic formulation) and an adequate interpolation procedure.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bauzet, C.: Time-splitting approximation of the Cauchy problem for a stochastic conservation law. Math. Comput. Simul. 118, 73–86 (2015)

    Article  MathSciNet  Google Scholar 

  2. Bauzet, C., Vallet, G., Wittbold, P.: The Cauchy problem for conservation laws with a multiplicative stochastic perturbation. J. Hyperb. Differ. Equ. 9(4), 661–709 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bauzet, C., Vallet, G., Wittbold, P.: The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation. J. Funct. Anal. 266(4), 2503–2545 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bauzet, C., Charrier, J., Gallouët, T.: Convergence of flux-splitting finite volume schemes for hyperbolic scalar conservation laws with a multiplicative stochastic perturbation. Math. Comput. 85(302), 2777–2813 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bauzet, C., Charrier, J., Gallouët, T.: Convergence of monotone finite volume schemes for hyperbolic scalar conservation laws with multiplicative noise. Stoch. Partial Differ. Equ. Anal. Comput. 4(1), 150–223 (2016)

    MathSciNet  MATH  Google Scholar 

  6. Chen, G.-Q., Ding, Q., Karlsen, K.H.: On nonlinear stochastic balance laws. Arch. Ration. Mech. Anal. 204(3), 707–743 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1992)

    Book  MATH  Google Scholar 

  8. Debussche, A., Vovelle, J.: Scalar conservation laws with stochastic forcing. J. Funct. Anal. 259(4), 1014–1042 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Dotti, S., Vovelle, J.: Convergence of approximations to stochastic scalar conservation laws. Arch. Ration. Mech. Anal. 230(2), 539–591 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  10. Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. In Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII, pages 713–1020. North-Holland, Amsterdam (2000)

  11. Feng, J., Nualart, D.: Stochastic scalar conservation laws. J. Funct. Anal. 255(2), 313–373 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gess, B., Souganidis, P.E.: Scalar conservation laws with multiple rough fluxes. Commun. Math. Sci. 13(6), 1569–1597 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gess, B., Perthame, B., Souganidis, P.E.: Semi-discretization for stochastic scalar conservation laws with multiple rough fluxes. SIAM J. Numer. Anal. 54(4), 2187–2209 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gess, B., Souganidis, P.E.: Long-time behavior, invariant measures and regularizing effects for stochastic scalar conservation laws. Comm. Pure Appl. Math. 70(8), 1562–1597 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hofmanová, M.: Scalar conservation laws with rough flux and stochastic forcing. Stochastic partial differential equations. Anal. Comput. 4(3), 635–690 (2016)

    MathSciNet  MATH  Google Scholar 

  16. Holden, H., Risebro, N. H.: A stochastic approach to conservation laws. In Third International Conference on Hyperbolic Problems, Vol. I, II (Uppsala, 1990), pages 575–587. Studentlitteratur, Lund (1991)

  17. Karlsen, K.H., Storrøsten, E.B.: On stochastic conservation laws and Malliavin calculus. J. Funct. Anal. 272(2), 421–497 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kim, Y.: Asymptotic behavior of solutions to scalar conservation laws and optimal convergence orders to \(N\)-waves. J. Differ. Equ. 192(1), 202–224 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. Koley, U., Majee, A.K., Vallet, G.: A finite difference scheme for conservation laws driven by Levy noise. IMA J. Numer. Anal. 38(2), 998–1050 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kröker, I., Rohde, C.: Finite volume schemes for hyperbolic balance laws with multiplicative noise. Appl. Numer. Math. 62(4), 441–456 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lions, P.-L., Perthame, B., Souganidis, P.E.: Stochastic averaging lemmas for kinetic equations. In Séminaire Laurent Schwartz—Équations aux dérivées partielles et applications. Année 2011–2012, Sémin. Équ. Dériv. Partielles, pages Exp. No. XXVI, 17. École Polytech., Palaiseau (2013)

  22. Lions, P.-L., Perthame, B., Tadmor, E.: A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Amer. Math. Soc. 7(1), 169–191 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lions, P.-L., Perthame, B., Tadmor, E.: Kinetic formulation of the isentropic gas dynamics and \(p\)-systems. Commun. Math. Phys. 163(2), 415–431 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lions, P.-L., Perthame, B., Souganidis, P.E.: Scalar conservation laws with rough (stochastic) fluxes. Stoch. Partial Differ. Equ. Anal. Comput. 1(4), 664–686 (2013)

    MathSciNet  MATH  Google Scholar 

  25. Lions, P.-L., Perthame, B., Souganidis, P.E.: Scalar conservation laws with rough (stochastic) fluxes: the spatially dependent case. Stoch. Partial Differ. Equ. Anal. Comput. 2(4), 517–538 (2014)

    MathSciNet  MATH  Google Scholar 

  26. Makridakis, C., Perthame, B.: Sharp CFL, discrete kinetic formulation, and entropic schemes for scalar conservation laws. SIAM J. Numer. Anal. 41(3), 1032–1051 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mohamed, K., Seaid, M., Zahri, M.: A finite volume method for scalar conservation laws with stochastic time-space dependent flux functions. J. Comput. Appl. Math. 237(1), 614–632 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Perthame, B.: Kinetic formulation of conservation laws, volume 21 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford (2002)

  29. Perthame, B.: Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure. J. Math. Pures Appl. (9) 77(10), 1055–1064 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  30. Schilling, R. L., Partzsch, L.: Brownian Motion. De Gruyter Graduate. De Gruyter, Berlin, second edition, (2014). An introduction to stochastic processes, With a chapter on simulation by Björn Böttcher

  31. Vallet, G., Wittbold, P.: On a stochastic first-order hyperbolic equation in a bounded domain. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12(4), 613–651 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Weinan, E., Khanin, K., Mazel, A., Sinai, Y.: Invariant measures for Burgers equation with stochastic forcing. Ann. Math. 151(3), 877–960 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. CEMOI, Faculté de droit et d’économie, 15 avenue René Cassin, BP 7151, 97715, Saint-Denis, La Réunion, France

    Sylvain Dotti

  2. UMPA, CNRS, ENS Lyon, ENS de Lyon site Monod 46, allée d’Italie, 69364, Lyon Cedex 07, France

    Julien Vovelle

Authors
  1. Sylvain Dotti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Julien Vovelle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Julien Vovelle.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Julien Vovelle was supported by ANR projects STOSYMAP and STAB.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dotti, S., Vovelle, J. Convergence of the Finite Volume method for scalar conservation laws with multiplicative noise: an approach by kinetic formulation. Stoch PDE: Anal Comp 8, 265–310 (2020). https://doi.org/10.1007/s40072-019-00146-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40072-019-00146-6

Keywords

Mathematics Subject Classification

Search

Navigation