A New Finite-Difference Scheme for Parabolic Differential Equations

Abstract

Finite-difference methods for unsteady heat conduction are considered. It is shown that the Crank-Nicolson scheme, although unconditionally stable in the mathematical sense, produces physically unrealistic solutions when the time step is large. The fully implicit scheme shows more satisfactory behavior, but is less accurate for small time steps. A new scheme is presented that is satisfactory for the entire range of time steps. It is also shown how the scheme can be applied to boundary-layer situations. The performance of the new scheme is demonstrated by its application to a heat conduction problem and a boundary-layer problem.