On a time-splitting method for a scalar conservation law with a multiplicative stochastic perturbation and numerical experiments

Abstract

In this paper, we present a numerical scheme for a first-order hyperbolic equation of nonlinear type perturbed by a multiplicative noise. The problem is set in a bounded domain D of \({\mathbb{R}^{d}}\) and with homogeneous Dirichlet boundary condition. Using a time-splitting method, we are able to show the existence of an approximate solution. The result of convergence of such a sequence is based on the work of Bauzet–Vallet–Wittbold (J Funct Anal, 2013), where the authors used the concept of measure-valued solution and Kruzhkov’s entropy formulation to show the existence and uniqueness of the stochastic weak entropy solution. Then, we propose numerical experiments by applying this scheme to the stochastic Burgers’ equation in the one-dimensional case.

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References

  1. 1.

    Balder, E.J.: Lectures on Young measure theory and its applications in economics. Rend. Istit. Mat. Univ. Trieste 31(suppl. 1), 1–69 (2000). Workshop on Measure Theory and Real Analysis (Italian) (Grado, 1997)

  2. 2.

    Bauzet C., Vallet G., Wittbold P.: The Cauchy problem for a conservation law with a multiplicative stochastic perturbation. Journal of Hyperbolic Differential Equations 9(4), 661–709 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Bauzet, C., Vallet, G., Wittbold, P.: The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation. Journal of Functional Analysis (2013)

  4. 4.

    Bensoussan A., Glowinski R., Raşcanu A.: Approximation of some stochastic differential equations by the splitting up method. Appl. Math. Optim. 25(1), 81–106 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    Carrillo J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147(4), 269–361 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

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

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992)

  8. 8.

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

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    E, W., Khanin, K., Mazel, A., Sinai, Y.: Invariant measures for Burgers equation with stochastic forcing. Ann. of Math. (2) 151(3), 877–960 (2000)

    Google Scholar 

  10. 10.

    Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992)

  11. 11.

    Eymard, R., Gallouët, T., Herbin, R.: Existence and uniqueness of the entropy solution to a nonlinear hyperbolic equation. Chinese Ann. Math. Ser. B 16(1), 1–14 (1995). A Chinese summary appears in Chinese Ann. Math. Ser. A 16 (1995), no. 1, 119

  12. 12.

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

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Gagneux, G., Madaune-Tort, M.: Analyse mathématique de modèles non linéaires de l’ingénierie pétrolière, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 22. Springer, Berlin (1996)

  14. 14.

    Hofmanová, M.: Bhatnagar-gross-krook approximation to stochastic scalar conservation laws (2013)

  15. 15.

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

  16. 16.

    Kloeden, P.E., Platen, E.: Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23. Springer, Berlin (1992)

  17. 17.

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

    Article  MATH  MathSciNet  Google Scholar 

  18. 18.

    LeVeque, R.J.: Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)

  19. 19.

    Málek, J., Nečas, J., Rokyta, M., Ruzicka, M.: Weak and measure-valued solutions to evolutionary PDEs, Applied Mathematics and Mathematical Computation, vol. 13. Chapman & Hall, London (1996)

  20. 20.

    Panov E.Y.: On measure-valued solutions of the Cauchy problem for a first-order quasilinear equation. Izv. Ross. Akad. Nauk Ser. Mat. 60(2), 107–148 (1996)

    Article  MathSciNet  Google Scholar 

  21. 21.

    Peyroutet, F.: Etude d’une méthode de splitting pour des lois de conservation scalaires avec terme de source. Ph.D. thesis, U.P.P.A. (1999)

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Correspondence to Caroline Bauzet.

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This research received support from the grant of the GENERAL COUNCIL of the Atlantic Pyrenees.

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Bauzet, C. On a time-splitting method for a scalar conservation law with a multiplicative stochastic perturbation and numerical experiments. J. Evol. Equ. 14, 333–356 (2014). https://doi.org/10.1007/s00028-013-0215-1

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Mathematics Subject Classification (2000)

  • 35L60
  • 60H15
  • 35L60

Keywords

  • Stochastic PDE
  • First-order hyperbolic equation
  • Dirichlet problem
  • Multiplicative stochastic perturbation
  • Young’s measures
  • Kruzhkov’s semi-entropy
  • Splitting method
  • Stochastic Burgers’ equation
  • Euler scheme
  • Godunov scheme