Analysis of Possibilities of Algorithms’ Selection and Adaptation for Differential Dynamic Models’ Numerical Implementation

• Date: 2020
• Main Authors: Верлань, Андрій, Положаєнко, Сергій
• Format: Article
• Language: Ukrainian
• Published: Кам'янець-Подільський національний університет імені Івана Огієнка 2020
• Online Access: http://mcm-math.kpnu.edu.ua/article/view/224857
• Journal Title: Mathematical and computer modelling. Series: Physical and mathematical sciences

The paper analyzes possibilities of selection and adaptation of computational algorithms in the implementation of differential dynamic models, i.e. in solving ordinary differential equations. The limited resources of computer-integrated systems determine the requirements for the computing and control systems’ performance, which indicates the urgency of target selection or adaptation of numerical methods for solving equations of dynamics objects. The process of improving numerical methods has a long history and does not stop until now. The growing complexity of the studied dynamic objects has led to the development of implicit methods for dynamics’ numerical analysis, but research shows that the use of implicit methods is justified, if we assume the use of a significant step of integrating the source system. In addition, it is found that the available results on the formalization of power methods for solving algebraic equations in the integration step and their adaptation in computer use are still insufficient to address their use in the study of complex dynamic objects. The integration step upper limit indicates the inexpediency of using high-order Runge-Kutta methods for the purposes of modeling the dynamics of the studied systems in real time. Accordingly, with regard to quadrature methods, the problem of formalizing the construction is not solved in general. Thus, the task of selecting the optimal method can be formulated as follows: to determine the numerical method for the modeled object’s dynamics equations’ integration, for which the required speed of the control system can be achieved, and the error of solving the dynamics equations does not exceed the specified value. The analysis of the properties of different groups of numerical methods is carried out, which makes it possible to conclude that in choosing the best method the initial set of the required methods should be formed based on the single-step methods of the Runge-Kutta type and the quadrature methods no higher than the fourth order. In implementing stationary modes, the initial group of methods should also include the multi-step methods — explicit and the «forecast-correction» type