Explicit numerical approximations for McKean-Vlasov stochastic differential equations in finite and infinite time

Inspired by the stochastic particle method, this paper establishes an easily implementable explicit numerical method for McKean-Vlasov stochastic differential equations (MV-SDEs) with superlinear growth coefficients. The paper establishes the theory on the propagation of chaos in the $L^{q}$ sense. The optimal {uniform-in-time} strong convergence rate $1/2$-order of the numerical solutions is obtained for the interacting particle system. Furthermore, it is proved that the numerical solutions capture the long-term dynamical behaviors of MV-SDEs precisely, including moment boundedness, stability, and ergodicity. Moreover, a unique numerical invariant probability measure is yielded, which converges to the underlying invariant probability measure of MV-SDEs in the $L^2$-Wasserstein distance. Finally, several numerical experiments are carried out to illustrate the main results.