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 ha