Analysis of a Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDEs

This paper approximates viscosity solutions of fully nonlinear second order parabolic PDEs by a narrow-stencil finite difference spatial-discretization paired with the forward Euler method for the time discretization. Using generalized monotonicity of the numerical scheme and an iteration technique, we prove the numerical scheme is stable in both the \(l^2\) $l^2$ -norm and the \(l^{\infty }\) $l^{\infty }$ -norm. Then under this stability, we establish the convergence of the proposed numerical scheme to the viscosity solution of the underlying fully nonlinear second order parabolic PDEs based on different regularities with respect to time and space. Finally, we report some numerical experiments.