The Maximum Principle and Some of Its Applications

Abstract

AbstractThe subject of this paper is the maximum principle and its application for investigating the stability and convergence of finite difference schemes. To some extent, this is a survey of the works on constructing and investigating certain new classes of monotone difference schemes. In this connection the maximum principle for the derivatives discussed in this paper is of fundamental importance. It is used as a basis for proving the coefficient stability of difference schemes in Banach spaces and the monotonicity of economical schemes of full approximation. New results on unconditional stability of monotone difference schemes with weights, conservative explicit-implicit schemes (staggered schemes), monotone schemes of second-order approximation in arbitrary domains, and monotone difference schemes for multidimensional elliptic equations with mixed derivatives are given.