  An iterative finite difference scheme for mean field games (MFGs) is
proposed. The target MFGs are derived from control problems for
multidimensional systems with advection terms. For such MFGs, linearization
using the Cole-Hopf transformation and iterative computation using fictitious
play are introduced. This leads to an implementation-friendly algorithm that
iteratively solves explicit schemes. The convergence properties of the proposed
scheme are mathematically proved by tracking the error of the variable through
iterations. Numerical calculations show that the proposed method works stably
for both one- and two-dimensional control problems.
