Maximum norm contractivity of discretization schemes for the heat equation (Q1192512)

A necessary and sufficient condition for maximum norm contractivity of a class of discretization methods is presented for solving the heat equation. Applying that result to the well-known Crank-Nicolson method a criterion is obtained which is less restrictive than the weakest presently available sufficient condition. A comparison is made with other stability criteria and sufficient conditions obtained by \textit{M. N. Spijker} [Numer. Math. 42, 271-290 (1983; Zbl 0504.65030) and Math. Comput. 45, 377-392 (1985; Zbl 0579.65092)]. The theoretical results are illustrated by means of several examples, among which there is construction of some optimal explicit schemes.