SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 886-924, January 2021.
This paper presents a new narrow-stencil finite difference method for approximating viscosity solutions of Hamilton--Jacobi--Bellman equations. The proposed finite difference scheme naturally extends the Lax--Friedrichs scheme for first order fully nonlinear PDEs to second order fully nonlinear PDEs which are approximated by Lax--Friedrichs-like numerical operators. The crux for constructing such a numerical operator is to introduce a stabilization term, which is called a “numerical moment” and corresponds to the numerical viscosity term in the original Lax--Friedrichs scheme for first order PDEs. It is proved that the proposed Lax--Friedrichs-like scheme has a unique solution and is stable in both the $\ell^2$-norm and the $\ell^\infty$-norm. Moreover, the convergence of the proposed finite difference scheme to the viscosity solution of the underlying Hamilton--Jacobi--Bellman equation is also established using a novel discrete comparison argument.

中文翻译：

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哈密​​顿-雅各比-贝尔曼方程近似粘度解的窄模板有限差分法。

SIAM数值分析学报，第59卷，第2期，第886-924页，2021年1月。
本文提出了一种新的窄模板有限差分法，用于近似Hamilton-Jacobi-Bellman方程的粘度解。拟议的有限差分方案自然地将一阶完全非线性PDE的Lax-Friedrichs方案扩展为由Lax-Friedrichs类数值算子近似的二阶完全非线性PDE。构造这样的数值算子的关键是引入一个稳定项，称为“数值矩”，它对应于原始Lax-Friedrichs方案中一阶PDE的数值粘度项。事实证明，提出的类似Lax-Friedrichs的方案具有唯一的解决方案，并且在$ \ ell ^ 2 $范数和$ \ ell ^ \ infty $范数中都是稳定的。而且，