论文标题
二阶平均现场游戏的Theta-Scheme的错误估计
Error estimates of a theta-scheme for second-order mean field games
论文作者
论文摘要
我们介绍并分析了一个新的有限差异方案，依靠Theta-Method来解决单调二阶平均野外游戏。这些游戏由Fokker-Planck和Hamilton-Jacobi-Bellman方程式组成。 theta方法用于离散扩散项：我们以隐式和显式项的凸组组合近似它们。相比之下，我们使用明确的中心方案作为一阶项。假设运行成本是强烈的凸起和常规的，我们首先证明了在CFL条件下的单调性和theta-Scheme的稳定性。利用连续问题解决方案的规律性，我们估计了theta-scheme的一致性误差。我们的主要结果是theta-scheme的订单$ \ MATHCAL {o}（h^r）$的收敛速率，其中$ h $是空间变量的步长，而（0,1）$ in（0,1）$与连续问题的解决方案的Hölder连续性有关。
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 Hölder continuity of the solution of the continuous problem and some of its derivatives.
