A new accurate finite difference scheme for Neumann (insulated) boundary condition of heat conduction

The Neumann (or insulated) boundary condition is often encountered in engineering applications. The conventional finite difference schemes are either first-order accurate or second-order accurate but need a ghost point outside the boundary. Compact finite difference schemes are difficult to apply for multi-dimensional cases or for cylindrical and spherical coordinate cases. In this study, we present a kind of new and accurate finite difference schemes for the Neumann (insulated) boundary condition in Cartesian, cylindrical, and spherical coordinates, respectively. Combined with the Crank–Nicholson finite difference method or other higher-order methods, the overall scheme is proved to be unconditionally stable and provides much more accurate numerical solutions. The numerical errors and convergence rates of the solution are tested by several examples. Results show that the new method is promising.