Weak and strong error analysis for mean-field rank-based particle approximations of one-dimensional viscous scalar conservation laws. (English) Zbl 1518.65017
Summary: In this paper, we analyse the rate of convergence of a system of \(N\) interacting particles with mean-field rank-based interaction in the drift coefficient and constant diffusion coefficient. We first adapt arguments by P. Kolli and M. Shkolnikov [Ann. Probab. 46, No. 2, 1042–1069 (2018; Zbl 1430.60057)] to check trajectorial propagation of chaos with optimal rate \({N^{-1/2}}\) to the associated stochastic differential equations nonlinear in the sense of McKean. We next relax the assumptions needed by M. Bossy [Math. Comput. 73, No. 246, 777–812 (2004; Zbl 1041.65015)] to check the convergence in \({L^1}(\mathbb{R})\) with rate \(\mathcal{O}(\frac{1}{\sqrt{N}}+h)\) of the empirical cumulative distribution function of the Euler discretization with step \(h\) of the particle system to the solution of a one-dimensional viscous scalar conservation law. Last, we prove that the bias of this stochastic particle method behaves as \(\mathcal{O}(\frac{1}{N}+h)\). We provide numerical results which confirm our theoretical estimates.
MSC:
| 65C35 | Stochastic particle methods |
| 60H35 | Computational methods for stochastic equations (aspects of stochastic analysis) |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 65C30 | Numerical solutions to stochastic differential and integral equations |
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