On the stability of certain difference schemes

The von Neumann stabil i ty criterion is employed in analyzing the stabili ty of a class of difference schemes for initial-value problems involving linear parabolic partial differential equations, u t = A u. I t is shown that , cont rary to the usual rule of thumb, there exist completely implicit difference schemes which are uncondit ionally unstable. Further , it is shown that the stabili ty properties of certain sets of corresponding schemes are closely related. We consider the linear parabolic partial differential equation for u = u ( m , t)