  Based on the assumption of the existence and uniqueness of the invariant
measure for McKean-Vlasov stochastic differential equations (MV-SDEs), a
self-interacting process that depends only on the current and historical
information of the solution is constructed for MV-SDEs. The convergence rate of
the weighted empirical measure of the self-interacting process and the
invariant measure of MV-SDEs is obtained in the W2-Wasserstein metric.
Furthermore, under the condition of linear growth, an EM scheme whose uniformly
1/2-order convergence rate with respect to time is obtained is constructed for
the self-interacting process. Then, the convergence rate between the weighted
empirical measure of the EM numerical solution of the self-interacting process
and the invariant measure of MV-SDEs is derived. Moreover, the convergence rate
between the averaged weighted empirical measure of the EM numerical solution of
the corresponding multi-particle system and the invariant measure of MV-SDEs in
the W2-Wasserstein metric is also given. In addition, the computational cost of
the two approximation methods is compared, which shows that the averaged
weighted empirical approximation of the particle system has a lower cost.
Finally, the theoretical results are validated through numerical experiments.
