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This paper presents a new narrow-stencil finite difference method forapproximating the viscosity solution of second order fully nonlinear ellipticpartial differential equations including Hamilton-Jacobi-Bellman equations. Theproposed finite difference method naturally extends the Lax-Friedrichs methodfor first order problems to second order problems by introducing a keystabilization and guiding term called a "numerical moment". The numericalmoment uses the difference of two (central) Hessian approximations to resolvethe potential low-regularity of viscosity solutions. It is proved that theproposed scheme is well posed (i.e, it has a unique solution) and stable inboth the l-2 norm and the l-infinity norm. The highlight of the paper is toprove the convergence of the proposed scheme to the viscosity solution of theunderlying fully nonlinear second order problem using a novel discretecomparison argument. This paper extends the one-dimensional analogous method ofFeng, Kao, and Lewis to the higher-dimensional setting. Numerical tests arepresented to gauge the performance of the proposed finite difference methodsand to validate the convergence result of the paper.

A Narrow-stencil finite difference method for approximating viscosity  solutions of fully nonlinear elliptic partial differential equations with  applications to Hamilton-Jacobi-Bellman equations