Runge–Kutta Pairs of Orders 5(4) Trained to Best Address Keplerian Type Orbits

The derivation of Runge–Kutta pairs of orders five and four that effectively uses six stages per step is considered. The coefficients provided by such a method are 27 and have to satisfy a system of 25 nonlinear equations. Traditionally, various solutions have been tried. Each of these solutions makes use of some simplified assumptions and offers different families of methods. Here, we make use of the most celebrated family to appear in the literature, where we may use as the last stage the first function evaluation from the next step (FSAL property). The family under consideration has the advantage of being solved explicitly. Actually, we arrive at a subsystem where all the coefficients are found with respect to five free parameters. These free parameters are adjusted (trained) in order to deliver a pair that outperforms other similar pairs of orders 5(4) in Keplerian type orbits, e.g., Kepler, perturbed Kepler, Arenstorf orbit or Pleiades. The training uses differential evolution technique. The finally proposed pair has a remarkable performance and offers on average more than a digit of accuracy in a variety of orbits

Sixth Order Numerov-Type Methods with Coefficients Trained to Perform Best on Problems with Oscillating Solutions

Numerov-type methods using four stages per step and sharing sixth algebraic order are considered. The coefficients of such methods are depended on two free parameters. For addressing problems with oscillatory solutions, we traditionally try to satisfy some specific properties such as reduce the phase-lag error, extend the interval of periodicity or even nullify the amplification. All of these latter properties come from a test problem that poses as a solution to an ideal trigonometric orbit. Here, we propose the training of the coefficients of the selected family of methods in a wide set of relevant problems. After performing this training using the differential evolution technique, we arrive at a certain method that outperforms the other ones from this family in an even wider set of oscillatory problems

Runge–Kutta Embedded Methods of Orders 8(7) for Use in Quadruple Precision Computations

High algebraic order Runge–Kutta embedded methods are commonly used when stringent tolerances are demanded. Traditionally, various criteria are satisfied while constructing these methods for application in double precision arithmetic. Firstly we try to keep the magnitude of the coefficients low, otherwise we may experience loss of accuracy; however, when working in quadruple precision we may admit larger coefficients. Then we are able to construct embedded methods of orders eight and seven (i.e., pairs of methods) with even smaller truncation errors. A new derived pair, as expected, is performing better than state-of-the-art pairs in a set of relevant problems

Sixth Order Numerov-Type Methods with Coefficients Trained to Perform Best on Problems with Oscillating Solutions

Numerov-type methods using four stages per step and sharing sixth algebraic order are considered. The coefficients of such methods are depended on two free parameters. For addressing problems with oscillatory solutions, we traditionally try to satisfy some specific properties such as reduce the phase-lag error, extend the interval of periodicity or even nullify the amplification. All of these latter properties come from a test problem that poses as a solution to an ideal trigonometric orbit. Here, we propose the training of the coefficients of the selected family of methods in a wide set of relevant problems. After performing this training using the differential evolution technique, we arrive at a certain method that outperforms the other ones from this family in an even wider set of oscillatory problems

Application of Intelligent and Digital Technologies to the Tasks of Wind Energy

The article considers the relevance and issues of wind turbine modeling, the principles of wind energy conversion (WEC) system operation, working areas and regulation. The influence of soft computing technologies on the different aspects of wind power systems, particularly in the fields of operation and maintenance, is considered. This article discusses the recent research, development and trends in soft computing techniques for wind-energy-conversion systems. For reliable analysis, the interaction of the wind-generator operation with the atmospheric boundary layer is considered. The authors give a detailed description of the approaches for the study and numerical modeling of the atmospheric boundary layer (ABL) in the vicinity of a wind farm. The study of the atmospheric boundary layer in the vicinity of the Ulyanovsk wind farm on the basis of cluster analysis of meteorological data is performed. Ten localizations of ABL homogeneous properties are identified. The subject of the study is the application of the results of cluster analysis to set linguistic variables in fuzzy inference algorithms as well as to adjust the initial conditions in the digital model of a wind generator. The results of cluster analysis made it possible to reasonably construct membership functions for the wind speed value in the fuzzy control algorithm to limit the output power of wind turbines. A simulation of the operation of a three-bladed horizontal type wind turbine for the conditions of one of the resulting clusters is performed, and the main regularities of the flow around the wind turbine are revealed. The results obtained are a valuable source for assessing the mutual influence of wind farms and the environment as well as wind farm site development

Runge–Kutta–Nyström Pairs of Orders 8(6) for Use in Quadruple Precision Computations

The second-order system of non-stiff Initial Value Problems (IVP) is considered and, in particular, the case where the first derivatives are absent. This kind of problem is interesting since since it arises in many significant problems, e.g., in Celestial mechanics. Runge–Kutta–Nyström (RKN) pairs are perhaps the most successful approaches for solving such type of IVPs. To achieve a pair attaining orders eight and six, we have to solve a well-defined set of equations with respect to the coefficients. Here, we provide a simplified form of these equations in a robust algorithm. When creating such pairings for use in double precision arithmetic, numerous conditions are often satisfied. First and foremost, we strive to keep the coefficients’ magnitudes small to prevent accuracy loss. We may, however, allow greater coefficients when working with quadruple precision. Then, we may build pairs of orders eight and six with significantly smaller truncation errors. In this paper, a novel pair is generated that, as predicted, outperforms state-of-the-art pairs of the same orders in a collection of important problems