This paper proposes a multiscale method for solving the numerical solution of mean field games which accelerates the convergence and addresses the problem of determining the initial guess. Starting from an approximate solution at the coarsest level, the method constructs approximations on successively finer grids via alternating sweeping, which not only allows for the use of classical time marching numerical schemes, but also enables applications to both local and nonlocal problems. At each level, numerical relaxation is used to stabilize the iterative process. A second-order discretization scheme is derived for higher order convergence. Numerical examples are provided to demonstrate the efficiency of the proposed method in both local and nonlocal, 1-dimensional and 2-dimensional cases.

本文提出了一种多尺度方法来求解均值场博弈的数值解，从而加快了收敛速度并解决了确定初始猜测的问题。该方法从最粗略的近似解开始，通过交替扫描在逐次精细的网格上构建近似，这不仅允许使用经典的时间行进数值方案，而且还可以应用于局部和非局部问题。在每个级别上，使用数值松弛来稳定迭代过程。导出了二阶离散化方案以实现更高阶的收敛性。数值例子说明了该方法在局部和非局部，一维和二维情况下的有效性。