Simulation of McKean Vlasov SDEs with super linear growth

We present two fully probabilistic numerical schemes, one explicit and one implicit, for the simulation of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts of super-linear growth and random initial condition. We provide a pathwise propagation of chaos result and show strong convergence for both schemes on the consequent particle system. The explicit scheme attains the standard $1/2$ rate in stepsize. From a technical point of view, we successfully use stopping times to prove the convergence of the implicit method although we avoid them altogether for the explicit one. The combination of particle interactions and random initial condition makes the proofs technically more involved. Numerical tests recover the theoretical convergence rates and illustrate a computational complexity advantage of the explicit over the implicit scheme. Comparative analysis is carried out on a stylized non Lipschitz MV-SDE and the neuron network model proposed in [J. Baladron et al., Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and FitzHugh-Nagumo neurons, The Journal of Mathematical Neuroscience, 2 (2012)]. We provide numerical tests illustrating \emph{particle corruption} effect where one single particle diverging can `corrupt' the whole system. Moreover, the more particles in the system the more likely this divergence is to occur.