In this paper, we study three numerical schemes for the McKean-Vlasov equation \[\begin{cases} \;dX_t=b(t, X_t, \mu_t) \, dt+\sigma(t, X_t, \mu_t) \, dB_t,\: \\ \;\forall\, t\in[0,T],\;\mu_t \text{ is the probability distribution of }X_t, \end{cases}\] where $X_0$ is a known random variable. Under the assumption on the Lipschitz continuity of the coefficients $b$ and $\sigma$, our first result proves the convergence rate of the particle method with respect to the Wasserstein distance, which extends a previous work [BT97] established in one-dimensional setting. In the second part, we present and analyse two quantization-based schemes, including the recursive quantization scheme (deterministic scheme) in the Vlasov setting, and the hybrid particle-quantization scheme (random scheme, inspired by the $K$-means clustering). Two examples are simulated at the end of this paper: Burger's equation and the network of FitzHugh-Nagumo neurons in dimension 3.

翻译：本文研究McKean-Vlasov方程的三种数值格式：\[\begin{cases} \;dX_t=b(t, X_t, \mu_t) \, dt+\sigma(t, X_t, \mu_t) \, dB_t,\: \\ \;\forall\, t\in[0,T],\;\mu_t \text{ 为 }X_t\text{ 的概率分布}, \end{cases}\] 其中$X_0$为已知随机变量。在系数$b$和$\sigma$满足Lipschitz连续的假设下，我们的第一个结果证明了粒子方法关于Wasserstein距离的收敛速率，推广了前期工作[BT97]在一维情形下的结论。第二部分给出并分析了两种基于量化的格式：Vlasov框架下的递归量化格式（确定性格式）以及受$K$-均值聚类启发的混合粒子-量化格式（随机格式）。本文最后通过两个算例进行模拟：Burger方程及三维FitzHugh-Nagumo神经元网络。