The article considers the algorithm of obtaining self-similar object, which forms by calculating the relative error of various finite-difference schemes for solving the second-order Cauchy problem using iterative process. The constructed graphical algorithm made it possible to simulate the image of the set for study, for example, to identify areas of stability for solving the problem. Using the program, it is possible to observe under what conditions and at what points the relative error value tends to infinity or remains in the area of certain values. The resulting model allows you to determine the nature of the changes in the set depending on the initial parameters, such as discretization step, estimation accuracy, areas of the complex plane. A computer graphical analysis of these phenomena is given. The computer can be turned into a kind of microscope and use it to observe the behavior of the boundaries of the region.