Universal computational algorithms and their justification for the approximate solution of differential equations

The paper is devoted to the problem that determines the typical characteristics of computing equipment associated with the amount of work needed to obtain a result at a given point in the computation domain. The use of grid methods is associated with the need for continuous processing and storage of data arrays determined by the number of grid elements, which is directly proportional to the performance of the systems used. We consider alternative approaches for the construction and justification of computational methods that are not focused on the grid structure of the approximations. The substantiation of the convergence of kinetic approximations to the solution of the Cauchy problem is obtained.