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

Particle method and quantization-based schemes for the simulation of the McKean-Vlasov equation