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.
Stability of delay parabolic difference equations
