Second order correct boundary conditions for the solution of parabolic partial differential equations

This paper is concerned with the numerical solution of the parabolic partial differential equation subject to boundary conditions of Neumann type. Douglas (2) and Rose (6) have shown that certain finite difference approximations, subject to first order correct boundary conditions, provide a globally first order correct solution to the differential system. In this paper it is shown that second order correct boundary condition can be used to obtain a globally second order correct solution. The boundary condition used involves an uncentered, second order correct difference approximation to the first derivative. No assumptions on the behavior of the solution of the differential equation outside the region of interest are made; such assumptions are necessary if a centered difference approximation is used