This book gives a systematic approach to the study of linear discrete parabolic problems. The concept of discrete time evolution equation is nowadays of great interest of investigation because it representss one way for the numerical solution of various continuous parabolic equations. This discrete time theory is on the one hand point the discrete analogue of the modern theory of parabolic differential equations in Banach space, but on the other hand some approaches posses distinctive features for which there is no analogue in the continuous case. Any modern discretization method used for nonstationary problems can be equivalently rewritten in the form of a single step method with respect to the time variable under certain reasonable restrictions. That is why the essential point of consideration is an analysis of first stage discrete problems.
General questions of stability of such problems within the framework of Banach spaces create the real core of the book especially its first part. Stability of the Cauchy problem for an evolution equation in discrete time for autonomous and nonautonomous cases is investigated. Then also problems with splitting operator and some more general equations, namely discrete evolution equations with a memory term are of great interest for the stability point of view. Part two is dedicated to the real numerical discretization methods for parabolic differential equations, namely Runge-Kutta methods. Stability estimates of the discrete solutions for different Runge-Kutta methods are derived. Then variable step size Runge-Kutta methods are studied and stability estimates for such methods are presented. In part three other discretizetion methods like the \(\theta\)-method and linear multistep methods are studied. Finally the stability of certain approximations to linear integro-differential equations in Banach space by applying the results of the previous chapters is discussed in the last part of the book.
Many results of this book are the collection of papers of Russian mathematicians who have worked in this field and have been published in editions almost unattainable for the readers outside of Russia.
Although the book is dealing only with linear problems its achievements are significant also for studying numerical methods for nonlinear parabolic equations. The main topic of the book is focused on problems of discretization abstract parabolic equations but there are also parts for example the problems with memory term and these results can be used also to parabolic partial differential and integro-differential equations.