Maximum norm contractivity of discretization schemes for the heat equation

A necessary and sufficient condition for maximum norm contractivity of a large class of discretization methods for solving the heat equations previously obtained is studied. Specializing this result for the well known Crank-Nicolson method the criterion delta t/(delta x) squared is less than or equal to 3/2, which is less restrictive than the weakest presently available sufficient condition delta t/(delta x) squared is less than or equal to 1 is arrived at. A comparison is made with other stability criteria and with sufficient conditions which are based on the concept of absolute monotonicity. The construction of some optimal explicit schemes are presented in the conclusion.