Rate of convergence of a particle method for the solution of a 1D viscous scalar conservation law in a bounded interval

• 成果类型: Article
• 署名作者: Bossy, M; Jourdain, B
• 署名单位: Inria; Institut Polytechnique de Paris; Ecole des Ponts ParisTech
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 1797-1832
• 关键词: STOCHASTIC DIFFERENTIAL-EQUATIONS; propagation; chaos

摘要

In this paper, we give a probabilistic interpretation of a viscous scalar conservation law in a bounded interval thanks to a nonlinear martingale problem. The underlying nonlinear stochastic process is reflected at the boundary to take into account the Dirichlet conditions. After proving uniqueness for the martingale problem, we show existence thanks to a propagation of chaos result. Indeed we exhibit a system of N interacting particles, the empirical measure of which converges to the unique solution of the martingale problem as N --> +infinity. As a consequence, the solution of the viscous conservation law can be approximated thanks to a numerical algorithm based on the simulation of the particle system. When this system is discretized in time thanks to the Euler-Upingle scheme [D. Upingle, Math. Comput. Simulation 38 (1995) 119-126], we show that the rate of convergence of the error is in O(Deltat + 1/rootN), where Deltat denotes the time step. Finally, we give numerical results which confirm this theoretical rate.