On the unconditional uniform stability of a finite difference scheme for the equation ut + ux = 0

Abstract

WHEN solving practical problems it is very advantageous to use finite difference schemes that are stable whatever the ratio between the mesh intervals. Such schemes are called unconditionally stable. In the present paper we prove the unconditional stability in C of a difference scheme for the equation u t + u x = 0. The method of proof could possibly be to other difference schemes. Necessary and sufficient conditions for the stability in C of difference schemes with constant coefficients are obtained in [1–4]. This answers the question of the stability in C for any fixed mesh interval ratio. But stability with any given fixed ratio does not in general guarantee unconditional stability in C. Two difference schemes for the equations of heat conduction are shown to be unconditionally uniformly stable in [5, 6]. The method given in the present paper is more general than that in [5, 6].