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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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)
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)
Bauzet, C., Vallet, G., Wittbold, P.: The Dirichlet problem for a conservation law with a multiplicative stochastic perturbation. Journal of Functional Analysis (2013)
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)
Carrillo J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147(4), 269–361 (1999)
Chen G.Q., Ding Q., Karlsen K.H.: On nonlinear stochastic balance laws. Arch. Ration. Mech. Anal. 204(3), 707–743 (2012)
Da Prato, G., Zabczyk, J.: Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications, vol. 44. Cambridge University Press, Cambridge (1992)
Debussche A., Vovelle J.: Scalar conservation laws with stochastic forcing. J. Funct. Anal. 259(4), 1014–1042 (2010)
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)
Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992)
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
Feng J., Nualart D.: Stochastic scalar conservation laws. J. Funct. Anal. 255(2), 313–373 (2008)
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)
Hofmanová, M.: Bhatnagar-gross-krook approximation to stochastic scalar conservation laws (2013)
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)
Kloeden, P.E., Platen, E.: Numerical solution of stochastic differential equations, Applications of Mathematics (New York), vol. 23. Springer, Berlin (1992)
Kröker I., Rohde C.: Finite volume schemes for hyperbolic balance laws with multiplicative noise. Appl. Numer. Math. 62(4), 441–456 (2012)
LeVeque, R.J.: Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)
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)
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)
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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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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DOI: https://doi.org/10.1007/s00028-013-0215-1