  We study the weak convergence behaviour of the Leimkuhler--Matthews method, a
non-Markovian Euler-type scheme with the same computational cost as the Euler
scheme, for the approximation of the stationary distribution of a
one-dimensional McKean--Vlasov Stochastic Differential Equation (MV-SDE). The
particular class under study is known as mean-field (overdamped) Langevin
equations (MFL). We provide weak and strong error results for the scheme in
both finite and infinite time. We work under a strong convexity assumption.
  Based on a careful analysis of the variation processes and the Kolmogorov
backward equation for the particle system associated with the MV-SDE, we show
that the method attains a higher-order approximation accuracy in the long-time
limit (of weak order convergence rate $3/2$) than the standard Euler method (of
weak order $1$). While we use an interacting particle system (IPS) to
approximate the MV-SDE, we show the convergence rate is independent of the
dimension of the IPS and this includes establishing uniform-in-time decay
estimates for moments of the IPS, the Kolmogorov backward equation and their
derivatives. The theoretical findings are supported by numerical tests.
