Maximum norm contractivity in the numerical solution of the one-dimensional heat equation

Abstract

In this paper we consider the one-dimensional heat conduction equation (Friedmann, 1964). To the numerical solution of the problem we apply the so-called (σ,θ)-method (Faragó, 1996; Thomée, 1990) which unites a few numerical methods. With the choice σ=0 we get the finite difference θ-method and the choice σ=16 results in the finite element method with linear elements. The most important question is the choice of the suitable mesh-parameters. The basic condition arises from the condition of the convergence (Faragó, 1996; Samarskii, 1977; Thomée, 1990). Further conditions can be obtained aiming at preserving some qualitative properties of the continuous problem. Some of them are the following: non-negativity, convexity, concavity, shape preservation and contractivity in some norms (Dekker and Verwer, 1984). Now we shall study the maximum norm contractivity. There are some results in the literature for the parameter choices which guarantee this property (Kraaijevanger, 1992; Samarskii, 1977; Thomée, 1990). However these papers specialize only on the finite difference methods and give sufficient conditions. We determine the necessary and sufficient conditions related to the (σ,θ)-method. We close the paper with numerical examples.