Monotone Mixed Finite Difference Scheme for Monge–Ampère Equation
Monotone Mixed Finite Difference Scheme for Monge–Ampère Equation
复制标题
Monge-Ampère方程的单调混合有限差分格式
DOI：
10.1007/s10915-018-0685-y
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发表时间：
2016
影响因子：
2.5
通讯作者：
Jessey Lin
中科院分区：
文献类型：
--
作者：
Yangang Chen;J. Wan;Jessey Lin
本文提出了一种求解二维Monge-Ampère方程的单调混合有限差分格式。为此，我们将Monge-Ampère方程转化为等价的Hamilton-Jacobi-Bellman(HJB)方程。在HJB格式的基础上，我们对单调性所在的网格点采用标准的7点模板离散化，并在其他地方采用半拉格朗日宽模板离散化，以确保整个计算域的单调性。通过将可允许控制集分成6个区域并对每个区域中的子问题进行优化，使优化问题在每个网格点上的计算量从标准7点模板离散化时的O(1)降低到O(1)，反之则降低到O(M)，其中离散化的控制集为$$M×M。证明了我们的数值格式满足一致性、稳定性、单调性和强比较原理，从而收敛于Monge-Ampère方程的粘性解。在数值结果中，当标准的7点模板离散化在整个计算域上单调应用时，可获得二阶收敛速度，否则可达到一阶收敛。与纯半拉格朗日宽模板格式相比，混合格式具有更小的离散化误差和更快的收敛速度。
In this paper, we propose a monotone mixed finite difference scheme for solving the two-dimensional Monge–Ampère equation. In order to accomplish this, we convert the Monge–Ampère equation to an equivalent Hamilton–Jacobi–Bellman (HJB) equation. Based on the HJB formulation, we apply the standard 7-point stencil discretization, which is second order accurate, to the grid points wherever monotonicity holds, and apply semi-Lagrangian wide stencil discretization elsewhere to ensure monotonicity on the entire computational domain. By dividing the admissible control set into six regions and optimizing the sub-problem in each region, the computational cost of the optimization problem at each grid point is reduced from $$O(M^2)$$O(M2) to O(1) when the standard 7-point stencil discretization is applied and to O(M) otherwise, where the discretized control set is $$M\times M$$M×M. We prove that our numerical scheme satisfies consistency, stability, monotonicity and strong comparison principle, and hence is convergent to the viscosity solution of the Monge–Ampère equation. In the numerical results, second order convergence rate is achieved when the standard 7-point stencil discretization is applied monotonically on the entire computation domain, and up to order one convergence is achieved otherwise. The proposed mixed scheme yields a smaller discretization error and a faster convergence rate compared to the pure semi-Lagrangian wide stencil scheme.