Universal Computational Algorithms and Their Justification for the Approximate Solution of Differential Equations

The problem of determining typical hardware characteristics associated with the amount of work needed to obtain a result at a given point in the computation domain is considered. Grid methods involve continuous processing and storage of data arrays, whose size is determined by the number of grid elements, which is directly proportional to the performance of the computer system used. Alternative approaches that do not rely on grid approximations are considered for constructing and justifying computational methods. The convergence of kinetic approximations to the solution of the Cauchy problem is substantiated.