Numerical Integration Of Differential Equations Research Articles

From low-rank retractions to dynamical low-rank approximation and back

In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well-established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a useful concept for other computational tasks on manifold as well, including interpolation tasks. In this work, we consider the application of retractions to the numerical integration of differential equations on fixed-rank matrix manifolds. This is closely related to dynamical low-rank approximation (DLRA) techniques. In fact, any retraction leads to a numerical integrator and, vice versa, certain DLRA techniques bear a direct relation with retractions. As an example for the latter, we introduce a new retraction, called KLS retraction, that is derived from the so-called unconventional integrator for DLRA. We also illustrate how retractions can be used to recover known DLRA techniques and to design new ones. In particular, this work introduces two novel numerical integration schemes that apply to differential equations on general manifolds: the accelerated forward Euler (AFE) method and the Projected Ralston–Hermite (PRH) method. Both methods build on retractions by using them as a tool for approximating curves on manifolds. The two methods are proven to have local truncation error of order three. Numerical experiments on classical DLRA examples highlight the advantages and shortcomings of these new methods.

Calculation of Rayleigh Damping Coefficients for a Transient Structural Analysis

Direct numerical integration of differential equations of motion is widely used by engineers to describe the behavior of structures under dynamic loading. The method entails directly integrating the motion equations over time. In the direct method, the damping matrix is formed as a linear combination of mass and stiffness matrices multiplied by the Rayleigh damping coefficients α and β, respectively. The Rayleigh damping coefficients have a significant effect on the response of building structures under dynamic loading. Therefore, the design values of the damping coefficients α and β have crucial importance to ensure accurate and reliable results in a dynamic analysis. The paper presents a time domain analysis for a building subjected to seismic excitations using the modal superposition and direct integration methods. The direct method considers the damping properties of building structures by Rayleigh damping coefficients obtained using various approaches. The building's response to seismic load is compared by response spectra. The authors proposed the least conservative approach for calculating the Rayleigh damping coefficients for analyzing a building in the time domain.

Modeling cable-pulley deployment systems of transformable rod structures

The aim of this article is to develop a simplified method for modeling cable-pulley deployment systems of rod structures based on the calculation of cable tensions and nodal driving forces with account for friction and other features of the system. Methods of theoretical mechanics, multibody dynamics, numerical integration of differential equations, and computer modeling were used during the research. The task of developing a simplified approach to modeling cable-pulley deployment systems for rod structures is considered. It is proposed to determine nodal driving forces by calculating cable tensions with account for friction and other features of the cable-pulley system, cables, and pulleys. To develop a model of cable-pulley deployment system, a rod system was chosen as the research object, which represents two sections of the transformable support truss of a reflector. Each section consists of diagonal and horizontal rods with tubular cross-sections. The sections are interconnected by hinge units. The structure is deployed using an upper and a lower cable, which pass through pulleys and are tensioned by an electric motor. The deploying forces are implemented by transferring the cable tension forces to the structure due to static friction and pressure between the cables and the pulleys. For further implementation of the model in an open-source software package, some simplifications were made due to the complexity of the design. A simplified method was developed for nodal driving force calculation in simulating rod structure deployment with the help of cables. The tensions, elongations, slacks, and neutral length of the cables and the forces transmitted from the cables to the pulleys were calculated as a function of time. Using them, the deployment of a rod structure was simulated for a constant cable speed. The results make it possible to control the rod system deployment time and rate depending on the characteristics and tension forces of the cables. The proposed approach is implemented using open-source software, and it provides modeling flexibility and reduces the model development and run time.

Temporal Subsampling Diminishes Small Spatial Scales in Recurrent Neural Network Emulators of Geophysical Turbulence

AbstractThe immense computational cost of traditional numerical weather and climate models has sparked the development of machine learning (ML) based emulators. Because ML methods benefit from long records of training data, it is common to use data sets that are temporally subsampled relative to the time steps required for the numerical integration of differential equations. Here, we investigate how this often overlooked processing step affects the quality of an emulator's predictions. We implement two ML architectures from a class of methods called reservoir computing: (a) a form of Nonlinear Vector Autoregression (NVAR), and (b) an Echo State Network (ESN). Despite their simplicity, it is well documented that these architectures excel at predicting low dimensional chaotic dynamics. We are therefore motivated to test these architectures in an idealized setting of predicting high dimensional geophysical turbulence as represented by Surface Quasi‐Geostrophic dynamics. In all cases, subsampling the training data consistently leads to an increased bias at small spatial scales that resembles numerical diffusion. Interestingly, the NVAR architecture becomes unstable when the temporal resolution is increased, indicating that the polynomial based interactions are insufficient at capturing the detailed nonlinearities of the turbulent flow. The ESN architecture is found to be more robust, suggesting a benefit to the more expensive but more general structure. Spectral errors are reduced by including a penalty on the kinetic energy density spectrum during training, although the subsampling related errors persist. Future work is warranted to understand how the temporal resolution of training data affects other ML architectures.

Ідентифікація параметрів математичних моделей та визначення діагностичних ознак

In this article, the object of research is two main units of the AR20N aggregate – electrohydraulic amplifier (EHA) and power cylinder (PC). Detection of the occurrence and development of faults in them will be considered on the basis of the identification of parameters of the developed mathematical models. The main objectives of the research were to identify the parameters of mathematical models of EHA and PC using different steps of time quantization, determination of the optimal step of time quantization for numerical integration of differential equations; determination of diagnostic features that have the most probable influence on the operation of aggregate units at the occurrence of various faults. For achievement of the set objectives the following tasks were solved: mathematical models of EHA, PC, and simulation model of the aggregate research on the test bench were developed; identification of parameters of mathematical models of EHA and PC by the method of least squares (MLS) was carried out; the permissible step of time quantization was determined; upper and lower permissible limits of the identifiable parameters for a serviceable aggregate were determined; a methodology for determining the diagnostic features that are most probable to determine the emerging faults of the aggregate has been developed. Methods are used for this purpose: analytical, numerical, statistical, hydraulic systems theory, and system identification theory. The following results were obtained: differential equations describing the displacements of the distributive spool (DS) of the EHA and the output link (OL) of the PC; algorithms for identifying the parameters of mathematical models by MLS; the permissible step of time quantization was determined; and a technique for determining the diagnostic features using the Shewhart diagnostic chart for the parameters of mathematical models was developed. The scientific and practical novelty of the obtained results consists of the following: mathematical models of EHA and PC of the aggregate have been developed; simulation model of the aggregate on the test bench has been developed; simulation of various faults of the aggregate units has been carried out and identification of parameters of mathematical models of these units has been carried out; changes of estimations of parameters of mathematical models of EHA and PC depending on changes of each influencing factor have been obtained; upper and lower permissible limits of parameters and average values of these parameters have been determined; diagnostic features that are most probable to influence aggregate performance have been determined. It is shown that time quantization step D t = 0.01 (c) provides the determination of DS and OL displacements with errors of 1.914×10-6mm and 2.157×10-5 mm.

NEUTRON STARS IN A UNIFORM DENSITY APPROXIMATION

Models of neutron stars are considered in the case of a uniform density distribution. A universal algebraic equation is obtained that is valid for any equation of state. This equation allows us to find the approximate mass of a star of a given density without resorting to the integration of differential equations. Solutions for various equations of state, including more realistic ones, are presented in the paper and differ from the exact solutions obtained by numerical integration of differential equations by no more than 20%.

Implementation of the Adams method for solving ordinary differential equations in the Sage computer algebra system

This work is devoted to the implementation and testing of the Adams method for solving ordinary differential equations in the Sage computer algebra system. The Sage computer algebra system has, to some extent, trivial means for numerical integration of ordinary differential equations, but at the same time, it is worth noting that this environment is convenient and practical for conducting computer experiments related to symbolic numerical calculations in it. The article presents the FDM package developed on the basis of the RUDN, which contains the developments of recent years, performed by M. D. Malykh and his students, for numerical integration of differential equations. In this package, attention is paid to the visualization of the calculation results, including the construction of various kinds of auxiliary diagrams, such as Richardson diagrams, as well as graphs of dependence, for example, the value of a function or step from a moment in time. The implementation of the Adams method will be considered from this package. In this article, this implementation of the Adams method will be tested on various examples of input data, and the method will also be compared with the Jacobi system. Exact and approximate values will be found and compared, and an estimate for the error will be obtained.

Stability and multistability of synchronization in networks of coupled phase oscillators

Coupled phase oscillators usually achieve synchronization as the coupling strength among oscillators is increased beyond a critical value. The stability of synchronous state remains an open issue. In this paper, we study the stability of the synchronous state in coupled phase oscillators. It is found that numerical integration of differential equations of coupled phase oscillators with a finite time step may induce desynchronization at strong couplings. The mechanism behind this instability is that numerical accumulated errors in simulations may trigger the loss of stability of the synchronous state. Desynchronization critical couplings are found to increase and diverge as a power law with decreasing the integral time step. Theoretical analysis supports the local stability of the synchronized state. Globally the emergence of synchronous state depends on the initial conditions. Other metastable ordered states such as twisted states can coexist with the synchronous mode. These twisted states keep locally stable on a sparse network but lose their stability when the network becomes dense.

Конвективный теплообмен и трение в тонком ламинарно-турбулентном пограничном слое на проницаемой поверхности затупленного конуса малого удлинения

Solving the problem of calculating convective heat transfer and friction in the laminar-turbulent boundary layer involves the need for numerical integration of differential equations, supplemented by those or other semiempirical models of apparent turbulent viscosity, which should be validated on the results of experimental studies, carried out in conditions, providing simulation of the gas dynamic picture of body flowing by a gas stream. Unfortunately, at present, there are no literature data on studies of laminar-turbulent heat exchange on the permeable surface of a blunted body, and under these conditions, one has to go by the way of comparing the calculated data with the results of experiments carried out on sharp bodies. Literature sources describe the results of studies carried out for a hemisphere, in which one of the semi-empirical models of apparent turbulent viscosity is used, tested on the results of experiments carried out on the impermeable surface of such shaped body. In this case, it was possible to obtain a physically consistent picture of the influence exerted by blowing gas through the wall on the degree of blocking of the convective heat flow. In the absence of qualitative experimental data on this issue, it seems reasonable to apply in practice the considered semi-empirical model of apparent turbulent viscosity to estimate the degree of blocking of convective heat flow and friction under the specified conditions. This article is devoted to the solution of a similar problem for the lateral surface of a blunted cone

Development of a mathematical model for assessing the dynamic quality of an assembled cutting tool structure

A software development methodology for calculating the static stiffness and dynamic characteristics of an assembled cutting tool by numerical integration of differential equations in the RKGS subroutine by the Runge—Kutta method is presented. This software can be used to assess the quality and design of assembled tools at the stage of developing a technological process for processing of materials. Keywords:assembled cutting tool, attachment point, static stiffness, dynamic characteristics, software. sigind1@yandex.ru, vladimir.malygin@yandex.ru, s.protasova@narfu.ru

Improving the Variational Quantum Eigensolver Using Variational Adiabatic Quantum Computing

The variational quantum eigensolver (VQE) is a hybrid quantum-classical algorithm for finding the minimum eigenvalue of a Hamiltonian that involves the optimization of a parameterized quantum circuit. Since the resulting optimization problem is in general nonconvex, the method can converge to suboptimal parameter values that do not yield the minimum eigenvalue. In this work, we address this shortcoming by adopting the concept of variational adiabatic quantum computing (VAQC) as a procedure to improve VQE. In VAQC, the ground state of a continuously parameterized Hamiltonian is approximated via a parameterized quantum circuit. We discuss some basic theory of VAQC to motivate the development of a hybrid quantum-classical homotopy continuation method. The proposed method has parallels with a predictor-corrector method for numerical integration of differential equations. While there are theoretical limitations to the procedure, we see in practice that VAQC can successfully find good initial circuit parameters to initialize VQE. We demonstrate this with two examples from quantum chemistry. Through these examples, we provide empirical evidence that VAQC, combined with other techniques (an adaptive termination criteria for the classical optimizer and a variance-based resampling method for the expectation evaluation), can provide more accurate solutions than “plain” VQE, for the same amount of effort.

Моделирование феррорезонансных режимов в нелинейных электрических цепях с учетом магнитного гистерезиса

Приведено решение задачи численного моделирования феррорезонансных режимов напряжений и токов в электрических цепях, возникающих при последовательном и параллельном соединении нелинейного индуктивного и емкостного элементов. Вебер-амперная характеристика индуктивного элемента задана с учетом гистерезиса. Известно, что феррорезонансные режимы возникают в электрических цепях, содержащих емкостные и нелинейные индуктивные элементы, в том числе силовые трансформаторы, дугогосящие реакторы, измерительные трансформаторы тока и напряжения, электродвигатели. Нелинейность характеристик индуктивных элементов служит причиной того, что при насыщении их магнитных систем происходит изменение дифференциальных индуктивностей, в результате возникают феррорезонансные, часто нештатные и даже аварийные режимы. Для имитационного моделирования магнитного гистерезиса использована модифицированная обратная модель Джайлса-Атертона. Численное интегрирование дифференциальных уравнений, описывающих динамические процессы в электрических цепях, выполнено явным методом Эйлера. Показано, что в результате феррорезонанса возникает тригеррный эффект, проявляющийся в форме скачкообразного изменения состояния цепи, вызванного неоднозначным характером вольт-амперной характеристики для входного напряжения и тока. Предложенный метод моделирования переходных и установившихся режимов в нелинейных электрических цепях с учетом магнитного гистерезиса может быть обобщен на случай произвольных цепей, в том числе, со сложной топологией.

MOVEMENT OF A PARTICLE ALONG AN INCLINED CYLINDER ROTATING AROUND ITS AXIS

It is well known that parts of agricultural machinery often have a cylindrical shape. This shape, for example, can be observed in the casing of lifting and transport machines, where the active working body rotates. Furthermore, drum grain dryers and triers use an inclined cylinder that rotates around its axis. In this case, the particles of the technological material interact with the rotating surface, which leads to their sliding, the nature of which depends on the value of the angle of inclination of the cylinder. In this study, the methods of differential geometry, vector algebra, theoretical mechanics, and numerical integration of differential equations consider the motion of a particle along the inner surface of an inclined cylinder rotating at a constant angular velocity around its axis. The axes of a fixed coordinate system are used to compose differential equations of motion. It was established that the proper initial conditions under which the particle would be stationary at a certain distance from the lower forming cylinder towards its rotation can be determined analytically. In case of movement along an inclined cylinder, the particle moves, among other things, in the axial direction, while reducing the amplitude of vibrations. Furthermore, it was found that the angle of inclination of the cylinder plays a significant role. If the latter is less than the angle of friction, then the vibrations stop, the movement of the particle stabilises, and it performs a rectilinear movement at a constant speed in the axial direction. If the angle of inclination of the cylinder is greater than or equal to the angle of friction, then the particle moves rapidly in the axial direction and its movement does not stabilise. The value of the angular velocity of rotation also plays a significant role. A certain amount of it provokes “sticking” of the particle, which does not depend on the inclination angle of the cylinder. The obtained analytical dependences can be used in the design of cylindrical working bodies of agricultural machines.

On symmetric-conjugate composition methods in the numerical integration of differential equations

We analyze composition methods with complex coefficients exhibiting the so-called “symmetry-conjugate” pattern in their distribution. In particular, we study their behavior with respect to preservation of qualitative properties when projected on the real axis and we compare them with the usual left-right palindromic compositions. New schemes within this family up to order 8 are proposed and their efficiency is tested on several examples. Our analysis shows that higher-order schemes are more efficient even when time step sizes are relatively large.

High order integrators obtained by linear combinations of symmetric-conjugate compositions

A new family of methods involving complex coefficients for the numerical integration of differential equations is presented and analyzed. They are constructed as linear combinations of symmetric-conjugate compositions obtained from a basic time-symmetric integrator of order 2n (n≥1). The new integrators are of order 2(n+k), k=1,2,…, and preserve time-symmetry up to order 4n+3 when applied to differential equations with real vector fields. If in addition the system is Hamiltonian and the basic scheme is symplectic, then they also preserve symplecticity up to order 4n+3. We show that these integrators are well suited for a parallel implementation, thus improving their efficiency. Methods up to order 10 based on a 4th-order integrator are built and tested in comparison with other standard procedures to increase the order of a basic scheme.

Generation and decay of counter-rotating vortices downstream of yawed wind turbines in the atmospheric boundary layer

Abstract

Numerical validation of blow-up solutions with quasi-homogeneous compactifications

We provide a numerical validation method of blow-up solutions for finite dimensional vector fields admitting asymptotic quasi-homogeneity at infinity. Our methodology is based on quasi-homogeneous compactifications containing quasi-parabolic-type and directional-type compactifications. Divergent solutions including blow-up solutions then correspond to global trajectories of associated vector fields with appropriate time-variable transformation tending to equilibria on invariant manifolds representing infinity. We combine standard methodology of rigorous numerical integration of differential equations with Lyapunov function validations around equilibria corresponding to divergent directions, which yields rigorous upper and lower bounds of blow-up time as well as rigorous profile enclosures of blow-up solutions.

Приближенная методика проектировочного баллистического расчета первых ступеней ракет-носителей

The initial stages of development of new types of launch vehicles involve performing the so-called design ballistic calculations. Currently, the relevance of design ballistics tasks is noticeably increasing in connection with the problems of deep space exploration and the creation of a new generation of space rocket systems, requiring significant financial investment. When developing design methods, motion models should not be inferior in accuracy to the uncertainties of the source data, and the algorithm of the technique should reflect the relationship of design parameters with the flight characteristics of the rocket. These principles formed the basis of the proposed methods for ballistic calculations of the active section of the trajectory of the first stages of heavy and super heavy launch vehicles. The technique is based on the results of the analysis of numerous trajectories corresponding to variations of various studied factors. The trajectory calculations were carried out using the method of numerical integration of differential equations of motion in a wide range of variation of design ballistic parameters and boundary conditions. The analysis made it possible to identify some general patterns of the controlled motion and to obtain the structure of the calculated dependencies in an analytical form.

Compositions of pseudo-symmetric integrators with complex coefficients for the numerical integration of differential equations

In this paper, we are concerned with the construction and analysis of a new class of methods obtained as double jump compositions with complex coefficients and projection on the real axis. It is shown in particular that the new integrators are symmetric and symplectic up to high orders if one uses a symmetric and symplectic basic method. In terms of efficiency, the aforementioned technique requires fewer stages than standard compositions of the same orders and is thus expected to lead to faster methods.

Effect of the magnus force and torque on the projectile flight range

The paper considers the origin of the Magnus force and torque in the course of the projectile flight; analytical dependencies for their determination are given and justified. The mutual orientation of the Magnus force and angular torque vectors with respect to those of the drag and lift forces, as well as with respect to the stalling, polar quenching and equatorial damping torques acting on the projectile in flight is discussed. The effect of the Magnus force and torque on the firing characteristics of artillery systems is assessed. A mathematical model of the projectile flight is presented, as well as constraints that are imposed on it and that do not considerably affect the accuracy of the description of the spatial motion of the projectile. The mathematical model is implemented programmatically on the basis of a standard subroutine for numerical integration of differential equations written in the Maple software and contains the motion equations for the projectile mass center, the motion equations with respect to the projectile mass center and equations that allow one to determine the coordinates of the projectile fall points in the base reference system. To assess the effect of Magnus force and torque on the projectile flight range, the difference method is employed, which consists in solving the system of differential equations of the spatial motion of the projectile such that changing the magnitude of the Magnus force and torque (at the assumption of the constancy of the aerodynamic forces and torques), one obtains the variation in the projectile flight range. A protocol and a scheme for assessing the effect of deviation of the projectile range on the accuracy of determination of the Magnus force strength and torque, as well as the results of numerical simulation of the deviation of the flight range for the ОФ-462Ж projectile of the 122-mm howitzer Д-30 depending on the accuracy of determination of aerodynamic force coefficients and Magnus torque are presented. It is shown that the largest deviations in the projectile flight range are observed when shooting at large launching angles and full charge (projectile speed 690 m / s); the smallest corresponding values are observed for the fourth charge (projectile speed 276 m / s) at small launching angles.