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.

翻译：我们研究了Leimkuhler-Matthews方法的弱收敛行为，该方法是一种与Euler格式计算成本相同的非马尔可夫Euler型格式，用于逼近一维McKean-Vlasov随机微分方程（MV-SDE）的平稳分布。所研究的特定类别被称为平均场（过阻尼）朗之万方程（MFL）。我们给出了该格式在有限时间和无限时间下的弱误差与强误差结果，并在强凸性假设下开展工作。基于对MV-SDE关联粒子系统的变分过程和Kolmogorov后向方程的细致分析，我们证明该方法在长时间极限下（弱收敛阶为$3/2$）比标准Euler方法（弱收敛阶为$1$）具有更高阶的逼近精度。尽管我们使用相互作用粒子系统（IPS）来逼近MV-SDE，但收敛速率与IPS的维数无关，这包括建立IPS矩、Kolmogorov后向方程及其导数的关于时间一致有界衰减估计。理论结果得到了数值实验的支持。