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

Abstract: This paper presents a new narrow-stencil finite difference method for approximating the viscosity solution of second order fully nonlinear elliptic partial differential equations including Hamilton-Jacobi-Bellman equations. The proposed finite difference method naturally extends the Lax-Friedrichs method for first order problems to second order problems by introducing a key stabilization and guiding term called a "numerical moment". The numerical moment uses the difference of two (central) Hessian approximations to resolve the potential low-regularity of viscosity solutions. It is proved that the proposed scheme is well posed (i.e, it has a unique solution) and stable in both the l-2 norm and the l-infinity norm. The highlight of the paper is to prove the convergence of the proposed scheme to the viscosity solution of the underlying fully nonlinear second order problem using a novel discrete comparison argument. This paper extends the one-dimensional analogous method of Feng, Kao, and Lewis to the higher-dimensional setting. Numerical tests are presented to gauge the performance of the proposed finite difference methods and to validate the convergence result of the paper.