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In this paper, we introduce adaptive Euler-Maruyama schemes for McKean-Vlasovstochastic differential equations (SDEs) assuming only a standard monotonicitycondition on the drift and diffusion coefficients but no global Lipschitzcontinuity in the state variable for either, while global Lipschitz continuityis required for the measure component only. We prove moment stability of thediscretised processes and a strong convergence rate of $1/2$. Several numericalexamples, centred around a mean-field model for FitzHugh-Nagumo neurons,illustrate that the standard uniform scheme fails and that the adaptiveapproach shows in most cases superior performance to tamed approximationschemes. In addition, we introduce and analyse an adaptive Milstein scheme fora certain sub-class of McKean-Vlasov SDEs with linear measure-dependence of thedrift.

An adaptive Euler-Maruyama scheme for McKean-Vlasov SDEs with super-linear growth and application to the mean-field FitzHugh-Nagumo model