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We introduce and analyze a new finite-difference scheme, relying on thetheta-method, for solving monotone second-order mean field games. These gamesconsist of a coupled system of the Fokker-Planck and theHamilton-Jacobi-Bellman equation. The theta-method is used for discretizing thediffusion terms: we approximate them with a convex combination of an implicitand an explicit term. On contrast, we use an explicit centered scheme for thefirst-order terms. Assuming that the running cost is strongly convex andregular, we first prove the monotonicity and the stability of our theta-scheme,under a CFL condition. Taking advantage of the regularity of the solution ofthe continuous problem, we estimate the consistency error of the theta-scheme.Our main result is a convergence rate of order $\mathcal{O}(h^r)$ for thetheta-scheme, where $h$ is the step length of the space variable and $r \in(0,1)$ is related to the H\"older continuity of the solution of the continuousproblem and some of its derivatives.

Error estimates of a theta-scheme for second-order mean field games