NUMERICAL SIMULATIONS OF THE PERIODIC INVISCID BURGERS EQUATION WITH STOCHASTIC FORCING

Emmanuel Audusse (1) , Sébastien Boyaval (2) , Yueyuan Gao (3) , Danielle Hilhorst (3)

1 LAGA - Laboratoire Analyse, Géométrie et Applications
2 LHSV - Laboratoire d'Hydraulique Saint-Venant / Saint-Venant laboratory for Hydraulics
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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 egularities 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.

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Emmanuel Audusse, Sébastien Boyaval, Yueyuan Gao, Danielle Hilhorst (Dir.). NUMERICAL SIMULATIONS OF THE PERIODIC INVISCID BURGERS EQUATION WITH STOCHASTIC FORCING. 48, pp.308-320, 2015, ESAIM: Proceedings and Surveys. ⟨hal-00962213v2⟩