| Abstract: |
| We present numerical methods for first-order stochastic partial differential equations. We start by numerical simulations in a one-dimensional torus for a first order Burgers equation forced by a stochastic source term with zero spatial integral. The stochastic source term is a white noise in time but spatially correlated. We apply a finite volume scheme combining the Godunov numerical flux with the Euler-Maruyama integrator in time together with a Monte-Carlo method. We numerically find that the empirical mean converges to the spatial-average of the deterministic initial condition as the time becomes infinite. The empirical variance also stabilizes for large time, towards a limit which depends on the space regularity and on the intensity of the noise. We then consider a linear transport equation in one space dimension with a linear gradient noise and a given initial condition. In some deterministic limit cases, there are known counterexamples showing the non-uniqueness of uniformly bounded solutions. In such a case, we numerically search for a selection criteria and perform numerical simulations with the purpose of simulating the passage from the stochastic case to the deterministic case. We will also present and compare simulation results of the stochastic transport equation in Ito's sense and the deterministic associated parabolic equation. |
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