An approach is treated for numerical integration of ordinary differential equations systems of the first order with choice of a computation scheme, ensuring the required local precision. The treatment is made on the basis of schemes of Runge-Kutta-Fehlberg type. Criteria are proposed as well as a method for the realization of the choice of an 'optimum' scheme. The effectiveness of the presented approach to problems in the field of satellite dynamics is illustrated by results from a numerical experiment. These results refer to a case when a satisfactory global stability of the solution for all treated cases is available. The effectiveness has been evaluated as good, especially when solving multi-variable problems in the sphere of simulation modelling.
Deep Dive into Integration of the Equation of the artificial Earths Satellites Motion with Selection of Runge-Kutta-Fehlberg Schemes of Optimum Precision Order.
An approach is treated for numerical integration of ordinary differential equations systems of the first order with choice of a computation scheme, ensuring the required local precision. The treatment is made on the basis of schemes of Runge-Kutta-Fehlberg type. Criteria are proposed as well as a method for the realization of the choice of an ‘optimum’ scheme. The effectiveness of the presented approach to problems in the field of satellite dynamics is illustrated by results from a numerical experiment. These results refer to a case when a satisfactory global stability of the solution for all treated cases is available. The effectiveness has been evaluated as good, especially when solving multi-variable problems in the sphere of simulation modelling.
Introduction. The practice of computing often demands the solution of ordinary differential equation systems (ODES) of the first order: which are applied without preliminary order lowering. As a rule they are more efficient in cases when they can be applied [3,4] but Nystrom's schemes [5] (which are the most popular) have a more restricted area of application since they require the right part to be independent from the first derivative of the dependent variable y.
In order to minimize the error and the computational expenses for each integration step, used to make the computations, major importance is attributed to the step size for numerical integration and the order of the computational schemes. Approaches exist for determination of optimum step size, which are possible both with the one-step and the multi-step methods. The change of the step in the one-step methods is easier while the multi-step methods require re-computation of the function derivative values in new points and produce complications [6]. The possibility to change the order (line) of the integration method in some multi-step methods [2,7] attracts attention.
Two basic final approaches exist in the numerical integration of ODES. The first one is connected with integration through optimal step selection. The optimal step size is determined on the basis of error assessment; in general, at smaller error the step increases and vice versa. Optimization of the computational expenses is achieved along with ensuring global stability of the numerical solution at the end of the integration interval.
The second approach requires finding the solution in equidistant values by the independent variable. The integration by a constant step, however, doesn’t always meet the requirement for sufficient local error, connected with the type of the functions on the right side of the equations, hence it can influence the solution stability.
There exists, however, a possibility during integration with optimal step to obtain the solution in desired points on the independent variable on the basis of interpolation. In addition -special methods exist, combining the numerical integration and interpolation, with which the solution is obtained in arbitrary points in a natural way with increased computational efficiency [8,9].
Different methods and programs-integrators of common differential equations are developed and their efficiency has been examined. As a result of their large number we’ll mention only a few, having in common with the evident one-step methods of the Runge-Kutta type [10,11,12,13]. Although the efficiency estimations show some advantages in behalf of one or another method and computer programs, when solving test problems there isn’t any certainty as for which method is the most suitable one.
A number of methodological groups exist for numerical solution of (1). The one-step methods of Runge-Kutta type [1] are characterised by adaptability and easy programme realization. Different computing schemes, corresponding to those methods are known. The advantages of the schemes of the Runge-Kutta-Fehlberg (RKF) type [2,3] are due to the fact that with minimum additional computations, two solutions are simultaneously obtained with different precision:
The difference between the two solutions y y i i -
gives the exact value of the main member of the local error ) h ( O 1 p+ for a scheme of the p -th order, on the basis of which the local error can be estimated. The classical RKF schemes were followed by later schemes with enhanced efficiency as well as by a higher order of the solution precision [16,17,18,19]. In cases of computations made on the basis of a scheme of the p -th order, by changing the integration step in definite limits, the necessary local precision is obtained and hence -a certain stability of the solution.
Formulation of the Problem. In the integration region along the independent variable x, the local error is a variable quantity and depends on step h, on the p -th order of the integration scheme and on the type of the functions on the right side of (1). The minimization of the local error by means of stepsize control is not the only possible. Practically, the minimization of the local error by a stepsize control is not always suitable. Instead, we can use an integration scheme of the lowest possible order, which provides the necessary local precision in integration with constant step. In this way the computations can turn out to be considerably less than if a scheme is used of an order, providing precision for the entire integration region along an independent variable. The issue for selection of optimal order of the integration schemes is dated far back [20,21]. This is possible even more in case when parallel integration of several ODES is necessary when the solutions have different character and are obtained with different local precision.
In the classical one-step methods of the Runge-Kutta type, the choice of a scheme with suffic
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