NUMERICAL SIMULATIONS OF THE PERIODIC INVISCID BURGERS EQUATION WITH STOCHASTIC FORCING
Résumé
We perform numerical simulations of the first order Burgers equation with a stochastic source term in the one-dimensional torus. To that purpose we apply a Finite-Volume scheme combining Godunov numerical flux with Euler-Maruyama integrator in time. We perform the numerical tests with different regularities of the source term in space, while it has the regularity of a white noise in time. Our computations exhibit features of the solution, in particular their large time behavior, for various regularities in space. The expectation always converges to the space-average of the initial function as the time tends to infinity in all cases (even when the regularity in space is rougher than what is covered by the existing large-time theories). Moreover, the variance stabilizes, at a value depending on the space regularity and on the intensity of the noise. We perform Monte Carlo simulations for which we visualize both statistical averages and single realizations.
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