Some questions from the general theory of difference schemes

One of the rapidly developing branches of modern mathematics is the the ory of difference schemes for the solution of the differential equations of mathe matical physics. Difference schemes are also widely used in the general theory of differential equations as an apparatus for proving existence theorems and investi gating the differential properties of solutions. But here one is primarily interested only in the asymptotic (for ft-*0) properties of the difference approximations. The theory of difference schemes has a number of special problems. In the final analysis, of greatest importance from the point of view of numerical analysis is the determination of algorithms permitting one to obtain a solution of a differential equation on an electronic computer with a prescribed accuracy in a finite number of operations. One encounters in this connection the question of the quality of an algorithm, i.e. the manner in which the accur acy of the algorithm depends on 1) the amount of information on the original problem, and 2) the amount of computation (viz. the machine time spent in solving the problem with a prescribed accuracy). Experience with computers has stimulated the formulation of a number of special (for the theory of difference methods) problems: 1) the determination of the achievable order of accuracy of difference schemes for various classes of problems, 2) the construction of schemes for the solution of a wide class of problems with a certain guaranteed accuracy, 3) the construction of schemes giving increased accuracy in narrower classes of problems, 4) the development of methods for investigating the stability and con vergence of difference schemes, 5) the formulation of general principles for con structing stable difference schemes and economizing the amount of computations (economical schemes), and others. In the present article we dwell only on a circle of questions connected with such fundamental notions of the theory of difference schemes as stability and approximation. The main purpose of the article is to show how the results of the general theory of difference schemes can be used to formulate principles for constructing