Geodesic
Numerical simulations of the inviscid burgers equation with periodic boundary conditions and stochastic forcing
Emmanuel Audusse1 ; Sébastien Boyaval2 ; Yueyuan Gao3 ; Danielle Hilhorst3
1 Université Paris 13, Institut Galilée, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France,
2 Université Paris Est, Laboratoire d’hydraulique Saint-Venant (Ecole des Ponts ParisTech – EDF R& D – CEREMA), 6 quai Watier, 78401 Chatou, France
3 Laboratoire de Mathématiques, CNRS et Université de Paris-Sud, 91405 Orsay Cedex, France,
ESAIM. Proceedings, Tome 48 (2015), pp. 308-320.

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We perform numerical simulations in the one-dimensional torus for the first order Burgers equation forced by a stochastic source term with zero spatial integral. We suppose that this source term is a white noise in time, and consider various regularities in space. For the numerical tests, we apply a finite volume scheme combining the Godunov numerical flux with the Euler-Maruyama integrator in time. Our Monte-Carlo simulations are analyzed in bounded time intervals as well as in the large time limit, for various regularities in space. The empirical mean always converges to the space-average of the (deterministic) initial condition as t → ∞, just as the solution of the deterministic problem without source term, even if the stochastic source term is very rough. The empirical variance also stablizes for large time, towards a limit which depends on the space regularity and on the intensity of the noise.
DOI : 10.1051/proc/201448014

Emmanuel Audusse 1 ; Sébastien Boyaval 2 ; Yueyuan Gao 3 ; Danielle Hilhorst 3

1 Université Paris 13, Institut Galilée, 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France,
2 Université Paris Est, Laboratoire d’hydraulique Saint-Venant (Ecole des Ponts ParisTech – EDF R& D – CEREMA), 6 quai Watier, 78401 Chatou, France
3 Laboratoire de Mathématiques, CNRS et Université de Paris-Sud, 91405 Orsay Cedex, France, 
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     title = {Numerical simulations of the inviscid burgers equation with periodic boundary conditions and stochastic forcing},
     journal = {ESAIM. Proceedings},
     pages = {308--320},
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Emmanuel Audusse; Sébastien Boyaval; Yueyuan Gao; Danielle Hilhorst. Numerical simulations of the inviscid burgers equation with periodic boundary conditions and stochastic forcing. ESAIM. Proceedings, Tome 48 (2015), pp. 308-320. doi : 10.1051/proc/201448014. http://geodesic.mathdoc.fr/articles/10.1051/proc/201448014/

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