We introduce and analyze a new finite-difference scheme, relying on the theta-method, for solving monotone second-order mean field games. These games consist of a coupled system of the Fokker-Planck and the Hamilton-Jacobi-Bellman equation. The theta-method is used for discretizing the diffusion terms: we approximate them with a convex combination of an implicit and an explicit term. On contrast, we use an explicit centered scheme for the first-order terms. Assuming that the running cost is strongly convex and regular, we first prove the monotonicity and the stability of our theta-scheme, under a CFL condition. Taking advantage of the regularity of the solution of the 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 the theta-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 continuous problem and some of its derivatives.

翻译：我们引入并分析了一种基于theta方法的新型有限差分格式，用于求解单调二阶平均场博弈。这类博弈由Fokker-Planck方程与Hamilton-Jacobi-Bellman方程构成的耦合系统组成。theta方法用于离散扩散项：我们通过隐式项与显式项的凸组合对其进行近似。相比之下，我们对一阶项采用显式中心格式。假设运行成本具有强凸性和正则性，我们首先在CFL条件下证明了theta格式的单调性和稳定性。利用连续问题解的正则性，我们估计了theta格式的一致性误差。我们的主要结果是theta格式具有$\mathcal{O}(h^r)$阶收敛速率，其中$h$为空间变量步长，$r \in (0,1)$与连续问题解及其某些导数的Hölder连续性相关。