In this paper we propose a high-order numerical scheme for time-dependent mean field games systems. The scheme, which is built by combining Lagrange-Galerkin and semi-Lagrangian techniques, is explicit, conservative, consistent, and stable for large time steps compared with the space steps. We provide a convergence analysis for the exactly integrated Lagrange-Galerkin scheme applied to the Fokker-Planck equation, and we propose an implementable version with inexact integration. Finally, we validate the convergence rate of the high order method proposed by numerical simulations of two Mean Field Games problems.

翻译：本文提出了一种针对时间依赖平均场博弈系统的高阶数值格式。该格式结合了拉格朗日-伽辽金与半拉格朗日技术，具有显式、守恒、相容的特性，且相较于空间步长，在时间步长较大时仍能保持稳定。我们对精确积分的拉格朗日-伽辽金格式应用于福克-普朗克方程进行了收敛性分析，并提出了一种采用非精确积分的可执行版本。最后，通过两个平均场博弈问题的数值模拟，验证了所提高阶方法的收敛阶。