On the Instability of the Crank Nicholson Formula Under Derivative Boundary Conditions

Abstract

Finite-difference solutions are considered for the heat conduction equation in one space dimension subject to general boundary conditions involving linear combinations of the function and its space derivative. It is shown that under such conditions, instability can often arise even although “stable” formulae of the Crank Nicholson type are used. In particular, the persistent error discussed by Parker and Crank (1964) is shown to be a weak case of this more serious instability.