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The paper considers the construction of adaptive methods based on the explicit Runge–Kutta stages. The coefficients of these methods are adjusted to the problem being solved, using component-wise estimates of the eigenvalues of the Jacobi matrix with the maximum absolute values. Such estimates can be easily obtained at the stages of the explicit method, which practically does not require additional calculations. The effect of computational errors and stiffness of the problem on the stability and accuracy of the numerical solution is studied. The analysis allows one to construct efficient explicit methods that are not inferior to implicit methods in solving many stiff problems. New nested pairs of adaptive methods are proposed, and the results of numerical experiments are presented.

The main computational costs of implicit Runge–Kutta methods are caused by solving a system of algebraic equations at every step. By introducing explicit stages, it is possible to increase the stage (or pseudo-stage) order of the method, which makes it possible to increase the accuracy and avoid reducing the order in solving stiff problems, without additional costs of solving algebraic equations. The paper presents implicit methods with an explicit first stage and one or two explicit internal stages. The results of solving test problems are compared with similar methods having no explicit internal stages.

We consider methods that are the inverse of the explicit Runge–Kutta methods. Such methods have some advantages, while their disadvantage is the low (first) stage order. This reduces the accuracy and the real order in solving stiff and differential-algebraic equations. New methods possessing properties of methods of a higher stage order are proposed. The results of the numerical experiments show that the proposed methods allow us to avoid reducing the order.

The solution of stiff problems is frequently accompanied by a phenomenon known as order reduction. The reduction in the actual order can be avoided by applying methods with a fairly high stage order, ideally coinciding with the classical order. However, the stage order sometimes fails to be increased; moreover, this is not possible for explicit and diagonally implicit Runge–Kutta methods. An alternative approach is proposed that yields an effect similar to an increase in the stage order. New implicit and stabilized explicit Runge–Kutta methods are constructed that preserve their order when applied to stiff problems.

Products formed in laser ablation of single-crystal silicon beneath a water layer under irradiation with nanosecond pulses from a fiber-optic ytterbium laser have been studied. SEM images of structures deposited by dewatering of the colloid solution formed in the course of the ablation are presented. IR spectroscopy and Raman spectroscopy were used to determine their chemical composition and structural features.

The features and basic functionality of the MVTU software package are considered. The software is intended for the research and design of a wide range of systems described by differential, algebraic, and difference equations. Examples of the solutions of the research and applied problems are given.

We consider the synthesis problem for low order controllers that provide given properties for a linear continuous system under uncertainty in object parameters. The synthesis is done with stability and quality criteria that have a simple dependence on the coefficients of the characteristic polynomial. We give examples of synthesizing controllers with given requirements to transition process quality, performance, and tracking accuracy.

We have studied the process of destruction of the surface of a single-crystalline silicon wafer scanned by the beam of a pulsed ytterbium-doped fiber laser radiation with a wavelength of λ = 1062 nm. It is established that the laser ablation can proceed without melting of silicon and the formation of a plasma plume. Under certain parameters of the process (radiation power, beam scan velocity, and beam overlap density), pronounced oxidation of silicon microparticles with the formation of a characteristic loose layer of fine powdered silicon dioxide has been observed for the first time. The range of lasing and beam scanning regimes in which the growth of SiO2 layer takes place is determined.

An implicit two-step Runge-Kutta method of fifth order is proposed for the numerical solution of differential and differential-algebraic equations. The location of nodes in this method makes it possible to estimate the values of higher derivatives at the initial and terminal points of an integration step. Consequently, the proposed method can be regarded as a finite-difference analog of the Obrechkoff method. Numerical results, some of which are presented in this paper, show that our method preserves its order while solving stiff equations and equations of indices two and three. This is the main advantage of the proposed method as compared with the available ones.

Efficient schemes for the implementation of diagonally implicit Runge-Kutta methods are considered. Methods of the 3rd and 4th orders are implemented. They are compared with known implicit solvers as applied to the solution of stiff and differential-algebraic equations.

Singly implicit diagonally extended Runge-Kutta methods make it possible to combine the merits of diagonally implicit methods (namely, the simplicity of implementation) and fully implicit ones (high stage order). Due to this combination, they can be very efficient at solving stiff and differential-algebraic problems. In this paper, fourth-order methods with an explicit first stage are examined. The methods have the third or fourth stage order. Consideration is given to an efficient implementation of these methods. The results of tests in which the proposed methods were compared with the fifth-order RADAU IIA method are presented.

Implementation schemes for second-order implicit Runge-Kutta methods are considered. The schemes allow one to reduce computational costs when solving stiff problems with low accuracy. The results of the comparison with implicit MATLAB solvers are presented.

Stiffly accurate Runge-Kutta collocation methods with explicit first stage are examined. The parameters of these methods are chosen so as to minimize the errors in the solutions to differential-algebraic equations of indices 2 and 3. This construction results in methods for solving such equations that are superior to the available Runge-Kutta methods.

In this paper, we discuss the elastic behavior of an isolated macromolecule in conjugated and non-conjugated ensembles during different modes of applying a mechanical force to the chain ends, namely, the mechanical effects corresponding to the methods of atomic force microscopy, magnetic levitation, and stretching of a macromolecule in an isotropic force field. Recently published results on the Langevin dynamics computer simulation of the stretching of a Gaussian polymer chain in different ensembles are analyzed. An analytical description of all the results of this simulation is given to show that the conclusion made by the authors of those studies about the nonequivalence of conjugated ensembles for Gaussian chains is not quite correct. A theoretical examination of the stretching of individual semirigid chains, which exhibit interesting behavior because of manipulations with actin molecules and microtubules, which are components of the cytoskeleton of most cells, is performed. We present rigorous analytical results about the stretching of chains composed of two or three freely jointed rods of equal length without consideration for excluded-volume inter-actions. It is shown that the stress-strain curves of these chains differ not only quantitatively but also qualitatively in different ensembles and that these curves for a chain of two freely jointed rods have an anomalous shape. These results on the different behavior of stress-strain for different modes of external mechanical impacts can be useful for interpreting experiments on the stretching of individual polymer molecules with different structures obtained via different methods, such as atomic-force microscopy and magnetic levitation.

When solving stiff problems the efficiency of the Runge-Kutta methods can be substantially improved if the parameters of the integration formula are adjusted to the problem at hand. The construction of such methods called adaptive is considered. The results obtained on test problems are compared with those obtained by known methods.

Explicit Runge-Kutta methods with the coefficients tuned to the problem of interest are examined. The tuning is based on estimates for the dominant eigenvalues of the Jacobian matrix obtained from the results of the preliminary stages. Test examples demonstrate that methods of this type can be efficient in solving stiff and oscillation problems.

Explicit Runge-Kutta methods with the stability domains extended along the real axis are examined. For these methods, a simple and efficient procedure for calculating the stability polynomials is proposed. Three techniques for constructing methods with given stability polynomials are considered. Methods of the second and third orders are constructed, and their accuracy as applied to solving the Prothero-Robinson equation is examined. A comparison of the above methods on some test problems is performed.

The simplest equations are considered that simulate the behavior of various error components of Runge-Kutta methods. The expressions for the local and global errors are obtained. The minimization of these errors allows one to construct explicit and implicit methods that have an improved accuracy when solving stiff and differential-algebraic problems.

Explicit multistep methods for solving Cauchy problems are examined. The proposed methods have their stability domains extended along the real axis and can be an alternative to one-step Runge-Kutta-Chebyshev methods when stiff problems are solved.

Diagonally implicit Runge-Kutta methods satisfying additional order conditions are examined. These conditions make it possible to solve differential algebraic equations of indices two and three to higher accuracy. Advantages of the proposed methods over other known techniques are demonstrated using test problems.

Explicit two-step Runge-Kutta methods with extended stability regions are considered as well as similar methods with an increased stage order. The advantage of two-step methods over the traditional one-step Runge-Kutta methods is shown.

The Runge-Kutta methods possessing the interpolation property, i.e., methods in which all coefficients belong to the interval [0, 1] are studied. Explicit and implicit methods of up to the fifth order inclusive that satisfy or almost satisfy the interpolation condition are considered.

A technique is proposed for constructing two-step Runge-Kutta methods on the basis of one-step methods. Explicit and diagonally implicit two-step methods with the second or third stage order are examined. Test problems are presented showing that the proposed methods are superior to conventional one-step techniques.

This review is concerned with the response of an isolated polymer chain subjected to the action of the two different modes of the mechanical impact on the chain ends. In one mode, the end-to-end distance is changed in a controlled fashion and the fluctuating response force is measured; in the second case, an external stress field is applied to the chain end, and the measured response of the system is the fluctuating end-to-end distance. The main attention is focused on the results of the computer-aided simulation experiments and theoretical results. Upon stretching of an ideal chain, a real chain in a good solvent, or a globule, the resultant strain-force and force-strain dependences are shown to be different for chains with finite length L; however, this difference diminishes with an increase in the length of a molecule. When the anchored Gaussian chain is separated from the adsorbing surface, this difference disappears in the limit of high L; however, in the neighborhood of the phase transition, some characteristics (fluctuations, distribution functions) appear to be critically different under different impact modes even in the thermodynamic limit. The example of an abnormal system is discussed: The behavior of a polymer chain compressed by a small piston is different in the conjugated ensembles, and, as the system increases in size, this difference becomes even more pronounced.

The reaction mixture from detoxication of technical-grade yperite with monoethanolamine was fractionated, and its composition was determined. The structures of the free thiomorpholine bases formed in the process were determined by gas chromatography—mass spectrometry.

An explicit multistep method of variable order for integrating stiff systems with high accuracy and low computational costs is examined. To stabilize the computational scheme, componentwise estimates are used for the eigenvalues of the Jacobian matrix having the greatest moduli. These estimates are obtained at preliminary stages of the integration step. Examples are given to demonstrate that, for certain stiff problems, the method proposed is as efficient as the best implicit methods.

Diagonally implicit Runge-Kutta methods are examined. It is shown that, for stiff problems, the methods based on the minimization of certain error functions have advantages over other methods; these functions are determined in terms of the errors for simplest model equations. Methods of orders three, four, five, and six are considered.