Convergence Rate for the Approximation of the Limit Law of Weakly Interacting Particles 2: Application to the Burgers Equation - Archive ouverte HAL Access content directly
Reports (Research Report) Year : 1994

## Convergence Rate for the Approximation of the Limit Law of Weakly Interacting Particles 2: Application to the Burgers Equation

Mireille Bossy
Denis Talay
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• PersonId : 833429

#### Abstract

In this paper, we construct a stochastic particles method for the Burgers equation with a monotonic initial condition; we prove that the convergence rate is $\displaystyleO\left(\frac1\sqrtN +\sqrt\D\right)$ for the $L^1(I\!\!R \times \Omega)$-norm of the error. To obtain that result, we link the PDE and the algorithm to a system of weakly interacting stochastic particles; the difficulty of the analysis comes from the discontinuity of the interaction kernel, equal to the Heaviside function. In~\citebossy_talay-93, we show how the algorithm and the result extend to the case of non monotonic initial conditions for the Burgers equation; we also treat the case of nonlinear PDE's related to particles systems with Lipschitz interaction kernels. Our next objective is to adapt our methodology to the (more difficult) case of the 2-D inviscid Navier-Stokes equation.

#### Domains

Computer Science [cs] Other [cs.OH]

### Dates and versions

inria-00074265 , version 1 (24-05-2006)

### Identifiers

• HAL Id : inria-00074265 , version 1

### Cite

Mireille Bossy, Denis Talay. Convergence Rate for the Approximation of the Limit Law of Weakly Interacting Particles 2: Application to the Burgers Equation. [Research Report] RR-2410, INRIA. 1994, pp.49. ⟨inria-00074265⟩

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