Different Orders Of Approximation Research Articles

Numerical parametrization of stationary axisymmetric black holes in a theory agnostic framework

The pursuit of a comprehensive theory of gravity has led to the exploration of various alternative models, necessitating a model-independent framework. The Konoplya-Rezzolla-Zhidenko (KRZ) parametrization offers a robust method for approximating stationary axisymmetric black hole spacetimes, characterized by a rapidly converging continued-fraction expansion. However, while analytical metrics benefit from this approach, numerical metrics derived from complex gravitational theories remain presenting computational challenges. Bridging this gap, we propose a method for a numerical KRZ parametrization, tested and demonstrated on pseudonumerical Kerr and Kerr-Sen spacetimes. Our approach involves constructing numerical grids to represent metric coefficients and using the grids for fitting the parameters up to an arbitrary order. We analyze the accuracy of our method across different orders of approximation, considering deviations in the metric functions and shadow images. In both Kerr and Kerr-Sen cases, we observe rapid convergence of errors with increasing orders of continued fractions, albeit with variations influenced by spin and charge. Our results underscore the potential of the proposed algorithm for parametrizing numerical metrics, offering a pathway for further investigations across diverse gravity theories. Published by the American Physical Society 2024

NONLINEAR EFFECTS IN VISCOELASTIC DROP SHAPE OSCILLATIONS

A study of axisymmetric shape oscillations of viscoelastic drops in a vacuum is conducted, using the method of weakly nonlinear analysis. The motivation is the relevance of the shape oscillations for transport processes across the drop surface, as well as fundamental interest. The study is performed for, but not limited to, the two-lobed mode of initial drop deformation. The Oldroyd-B model is used for characterizing the liquid rheological behavior. The method applied yields a set of governing equations, as well as boundary and initial conditions, for different orders of approximation. In the present paper, the equations and solutions up to second order are presented, together with the characteristic equation for the viscoelastic drop. The characteristic equation has an infinite number of roots, which determine the time dependency of the oscillations. Solutions of the characteristic equation are validated against experiments on acoustically levitated individual viscoelastic aqueous polymer solution drops. Experimental data consist of decay rate and oscillation frequency of free damped drop shape oscillations. With these data, solutions of the characteristic equation dominating the oscillations are identified. The theoretical analysis reveals nonlinear effects, such as the excess time in the prolate shape and frequency change for varying initial deformation amplitude. The influences of elasticity, measured by the stress relaxation and deformation retardation time scales, are quantified, and the effects are compared to the Newtonian case in the moderate-amplitude regime.

Efficient numerical evaluation of weak restricted compositions

We propose an algorithm to calculate the number of weak compositions, wherein each part is restricted to a different range of integers. This algorithm performs different orders of approximation up to the exact solution by using the Inclusion-Exclusion Principle. The great advantage of it with respect to the classical generating function technique is that the calculation is exponentially faster as the size of the numbers involved increases.

Theory-guided physics-informed neural networks for boundary layer problems with singular perturbation

Physics-informed neural networks (PINNs) are a recent trend in scientific machine learning research and modeling of differential equations. Despite progress in PINN research, large gradients and highly nonlinear patterns remain challenging to model. Thin boundary layer problems are prominent examples of large gradients that commonly arise in transport problems. In this study, boundary-layer PINN (BL-PINN) is proposed to enable a solution to thin boundary layers by considering them as a singular perturbation problem. Inspired by the classical perturbation theory and asymptotic expansions, BL-PINN is designed to replicate the procedure in singular perturbation theory. Namely, different parallel PINN networks are defined to represent different orders of approximation to the boundary layer problem in the inner and outer regions. In different benchmark problems (forward and inverse), BL-PINN shows superior performance compared to the traditional PINN approach and is able to produce accurate results, whereas the classical PINN approach could not provide meaningful solutions. BL-PINN also demonstrates significantly better results compared to other extensions of PINN such as the extended PINN (XPINN) approach. The natural incorporation of the perturbation parameter in BL-PINN provides the opportunity to evaluate parametric solutions without the need for retraining. BL-PINN demonstrates an example of how classical mathematical theory could be used to guide the design of deep neural networks for solving challenging problems.

Comparative analysis of different FOPI approximations and number of terms used on simulations of a battery-powered, field-oriented induction motor based electric vehicle traction system

Because of their enhanced performance, the fractional order proportional-integral (FOPI) controllers are becoming an appealing choice for controlling induction motor speed. To implement FOPI controllers, several fractional order integral approximations are available in the literature. The approximation used, and the order of approximation affects the speed tracking, transient response, and induction motor power consumption. This further affects the energy consumption analysis if simulations are conducted based on such approximations. In this paper an electric vehicle (EV) traction system is simulated to investigate the effect of such approximations on the simulations of a battery powered, induction motor driven EV system. The system consists of an indirect field-oriented induction motor, a lithium-ion battery bank, and a three-phase inverter. This work presents a quantitative analysis of the performance of FOPI controllers using different approximations, and order of approximations is presented. The controllers are evaluated based on speed tracking, transient response, computational time, and power consumption. Both step functions and standard drive cycles are used as the speed reference signal to evaluate the effects of using different approximations and different orders of approximation, when different references are used. This work establishes a reference set of simulations that can be used to infer the amount of error in battery state of charge, and state of health analysis conducted on such an EV system, when dealing with FOPI controllers under different approximations and related settings.

Polynomial mode approximations for longitudinal wave dispersion in solid and hollow circular cylinders

The characteristic equation for wave dispersion in solid or hollow circular cylinders is a transcendental equation: no closed-form solution exists and it can only be solved numerically. The characteristic equation for a hollow circular cylinder is substantially longer and more complex to solve than the equation for a solid cylinder. This paper examines, at different orders of approximation, two polynomial approximations (Jacobi mode approximation and an approximation by Widehammar, Gradin and Lundberg) which simplify the computation of dispersion curves. We focus on the approximation error for the first four modes of axially symmetric longitudinal waves, for both solid and hollow circular cylinders. The two polynomial approximations yield accurate dispersion curves for a solid cylinder, provided a sufficient order is used. For a hollow circular cylinder, the numerical limits of thinner cylinders prevent the use of the highest orders of approximation, limiting the accuracy of the dispersion curves. Classic theories for thin and thick cylindrical shells (by Naghdi and Cooper, and Herrmann and Mirsky), remain useful alternatives when computing dispersion curves for the first propagation mode for very thin hollow circular cylinders. We find that Widehammar’s approximation yields more accurate dispersion curves than Jacobi mode approximation for hollow cylinders, for a given order of approximation, as well as being more straightforward to program.

Effect of Delay Approximation of various orders on the open loop response of a Delayed Process Model

In present analysis, response of a selected delayed first order process model is compared for various orders of approximation of dead time present in the transfer function model. Effect of different orders of approximation of dead time is analysed on the open loop step response of the system in terms of percentage change in original and dead time approximated system models. For this purpose, consistency parameter in a headbox of a paper machine is selected as a process model as it closely approximates to a first order delayed process model.

A comparative study between the IGN-2 equations and the fully nonlinear, weakly dispersive Boussinesq equations

This work focuses on the comparison between the fully nonlinear, and weakly dispersive Boussinesq equations and the irrotational Green–Naghdi Level 2 (IGN-2) equations for incompressible and inviscid fluid and irrotational flow resulting from waves in finite water depth. We provide in this paper for the first time a comprehensive comparison of the GN/IGN and Boussinesq wave theories for strongly nonlinear and weakly dispersive waves. The Boussinesq approach results in many different theories depending on the different orders of approximation of nonlinearity and dispersion used in the corresponding perturbation series. The fully nonlinear, weakly dispersive Boussinesq equations have been widely used in problems involving coastal regions and harbors because it is more efficient than fully nonlinear, strongly dispersive Boussinesq equations. A competing theory is the strongly nonlinear IGN-2 wave theory. Since both sets of equations have comparable complexity, it is appropriate to compare the results of these equations with known numerically exact solutions to determine the advantages of each approach. In this study, we use the same iterative algorithm to obtain the steady solutions of the IGN-2 and Boussinesq equations for periodic waves. The steady solutions of the IGN-2 and Boussinesq equations are studied for four different wavelengths that cover a range of applicability of the theories. We show in this study that the Boussinesq equations give more accurate results for the velocity distribution at the wave-crest transection. The IGN-2 equations give more accurate results for the velocity distribution at the wave-trough transection. Moreover, it is shown that the IGN-2 equations give more accurate results on the wave speed and wave profile, and that they can treat accurately waves higher than the ones that the Boussinesq equations can treat. The calculations show that both of these sets of equations are very good for strongly nonlinear wave simulations and the limitations of each are different.

Possible existence of van der Waals macrodimers

ABSTRACTFew-body interactions offer the opportunity to study the isolated atom to few-body coupled molecules, and to condensed matter transitions. Atoms in molecules and in condensed matters are coupled by different orders of multipole–multipole interactions, which all stem from different orders of approximations from coulomb interactions between multiple charges. The lowest order multipole–multipole interaction is the dipole–dipole interaction, which is proportional to the size of the dipole. In this article, we use Rydberg atoms, which have more than 1000 times greater electric dipoles than the ground state atoms, to study the van der Waals interaction between few bodies. In addition to the large dipoles, the kinetic energy of the atoms is significantly reduced by reducing the temperature, which makes these interactions stable and observable. Here, we report on the 2D and 3D few-body interaction potentials and possible ways of creating semistable molecules in such an ultracold Rydberg gas with a temperature of ≈100 nK. Although we use Rydberg atoms in this article, this calculation can be applied to other states too. The results reported here are useful for studying repulsive van der Waals interactions and creating ultracold molecules.

Variational Integrators for Thermo‐Viscoelastic Discrete Systems

AbstractVariational integrators are modern time‐integration schemes based on a discretization of the underlying variational principle. In this paper, Hamilton's principle is approximated by an action sum, whose vanishing variation results in discrete Euler‐Lagrange equations or, equivalently, in discrete evolution equations for the position and momentum. In order to include the viscous and thermal virtual work (mechanical and thermal virtual dissipation), Hamilton's principle is extended by D'Alembert terms, which account for the time evolution equation of the internal variable and Fourier's law.From this variational formulation, variational integrators using different orders of approximation of the state variables as well as of the quadrature of the action integral are constructed and compared. A thermo‐viscoelastic double pendulum comprised of two discrete masses connected by generalized Maxwell elements, and subject to heat conduction between them serves as a discrete model problem. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A simple model of wave–current interaction

The interaction between a steady current and propagating surface waves is investigated by means of a perturbation approach, which assumes small values of the wave steepness and considers current velocities of the same order of magnitude as the amplitude of the velocity oscillations induced by wave propagation. The problems, which are obtained at the different orders of approximation, are characterized by a further parameter which is the ratio between the thickness of the bottom boundary layer and the length of the waves and turns out to be even smaller than the wave steepness. However, the solution is determined from the bottom up to the free surface, without the need to split the fluid domain into a core region and viscous boundary layers. Moreover, the procedure, which is employed to solve the problems at the different orders of approximation, reduces them to one-dimensional problems. Therefore, the solution for arbitrary angles between the direction of the steady current and that of wave propagation can be easily obtained. The theoretical results are compared with experimental measurements; the fair agreement found between the model results and the laboratory measurements supports the model findings.

Hybrid perturbation methods based on statistical time series models

In this work we present a new methodology for orbit propagation, the hybrid perturbation theory, based on the combination of an integration method and a prediction technique. The former, which can be a numerical, analytical or semianalytical theory, generates an initial approximation that contains some inaccuracies derived from the fact that, in order to simplify the expressions and subsequent computations, not all the involved forces are taken into account and only low-order terms are considered, not to mention the fact that mathematical models of perturbations not always reproduce physical phenomena with absolute precision. The prediction technique, which can be based on either statistical time series models or computational intelligence methods, is aimed at modelling and reproducing missing dynamics in the previously integrated approximation. This combination results in the precision improvement of conventional numerical, analytical and semianalytical theories for determining the position and velocity of any artificial satellite or space debris object. In order to validate this methodology, we present a family of three hybrid orbit propagators formed by the combination of three different orders of approximation of an analytical theory and a statistical time series model, and analyse their capability to process the effect produced by the flattening of the Earth. The three considered analytical components are the integration of the Kepler problem, a first-order and a second-order analytical theories, whereas the prediction technique is the same in the three cases, namely an additive Holt–Winters method.

Numerical integration on GPUs for higher order finite elements

The paper considers the problem of implementation on graphics processors of numerical integration routines for higher order finite element approximations. The design of suitable GPU kernels is investigated in the context of general purpose integration procedures, as well as particular example applications. The most important characteristic of the problem investigated is the large variation of required processor and memory resources associated with different degrees of approximating polynomials. The questions that we try to answer are whether it is possible to design a single integration kernel for different GPUs and different orders of approximation and what performance can be expected in such a case.

Optimal homotopy analysis method for nonlinear differential equations in the boundary layer

Optimal homotopy analysis method is a powerful tool for nonlinear differential equations. In this method, the convergence of the series solutions is controlled by one or more parameters which can be determined by minimizing a certain function. There are several approaches to determine the optimal values of these parameters, which can be divided into two categories, i.e. global optimization approach and step-by-step optimization approach. In the global optimization approach, all the parameters are optimized simultaneously at the last order of approximation. However, this process leads to a system of coupled, nonlinear algebraic equations in multiple variables which are very difficult to solve. In the step-by-step approach, the optimal values of these parameters are determined sequentially, that is, they are determined one by one at different orders of approximation. In this way, the computational efficiency is significantly improved, especially when high order of approximation is needed. In this paper, we provide extensive examples arising in similarity and non-similarity boundary layer theory to investigate the performance of these approaches. The results reveal that with the step-by-step approach, convergent solutions of high order of approximation can be obtained within much less CPU time, compared with the global approach and the traditional HAM.

Analytical approach to dynamic and vibration analysis of a spherical ball under contact stress

This paper presents a nonlinear model to illustrate the effect of contact stress in the vibration behavior of mechanical components, especially rotating systems. The problem is considered in the case of the vertical vibration of a sphere on a plate. The Hertzian contact theory is used to obtain the relationship between contact force and the deflection of the mass center of the sphere. Modeling the system by a mass and a nonlinear spring, the vibration equation of the mass center of the sphere is derived. The method of Lindstedt–Poincaré is implemented to solve the equation of motion, and obtain vibration characteristics under a compressive preload. The dependency of frequency on several parameters, such as initial applied force, initial amplitude of oscillation and the diameter of the sphere, is distinguished. Results show that increasing the initial applied force or the diameter of the ball raises the frequency, while increasing oscillation amplitude has an inverse effect. Finally, the accuracy and convergence of the solution are illustrated by comparison between different orders of approximation. Also, results are in good agreement with those extracted from numerical modeling.

The geometric stricture of the parameter space of the three-stage symplectic Runge-Kutta methods

The geometric structure of the parameter space of the three-stage symplectic Runge-Kutta methods is established. The elements of the structure are geometric places of points corresponding to the methods of different orders of approximation.

Design of a Discrete Flux Observer by the Power Series Approximation

The power series approximation method is proposed for real time implementations of a discrete flux observer. The proposed method improves the performance of the discrete flux observer in the case of a low sampling rate and high speed range, where the simple discrete flux observer converted by the Euler method cannot estimate the actual flux precisely. The performance of discrete flux observers with different orders of approximation is compared to find out the proper order of approximation. The validity of the proposed method is verified through simulation and experiment.

Intercepts of the momentum correlation functions in the $ \mu$ -Bose gas model and their asymptotics

The so-called \mu-deformed oscillator (or \mu-oscillator) introduced by A.Jannussis, though possesses rather exotic properties with respect to other better known deformed oscillator models, also has good potential for diverse physical applications. In this paper, the corresponding \mu-Bose gas model based on \mu-oscillators is explored. Within this model, the intercepts \lambda^(2)(K) and \lambda^(3)(K) of two- and three-particle momentum correlation functions are calculated with the goal of possible application for modeling the non-Bose type behavior of the intercepts of two- and three-pion correlations, observed in the experiments on relativistic heavy ion collisions. In derivation of intercepts, different orders of approximation in the deformation parameter \mu are considered. Our analysis also yields asymptotical behavior of the intercepts \lambda^(2)(K) and \lambda^(3)(K). The results for \lambda^(2)(K) are compared with experimental data, and with earlier known results drawn using other deformed Bose gas models.

Finite difference methods for second order in space, first order in time hyperbolic systems and the linear shifted wave equation as a model problem in numerical relativity

Motivated by the problem of solving the Einstein equations, we discuss high order finite difference discretizations of first order in time, second order in space hyperbolic systems.Particular attention is paid to the case when first order derivatives that can be identified with advection terms are approximated with non-centered finite difference operators.We first derive general properties of these discrete operators, then we extend a known result on numerical stability for such systems to general order of accuracy.As an application we analyze the shifted wave equation, including the behavior of the numerical phase and group speeds at different orders of approximations. Special attention is paid to when the use of off-centered schemes improves the accuracy over the centered schemes.

Higher order corrections for shallow-water solitary waves: elementary derivation and experiments

We present an elementary method to obtain the equations of the shallow-water solitary waves in different orders of approximation. The first two of these equations are solved to get the shapes and propagation velocities of the corresponding solitary waves. The first-order equation is shown to be equivalent to the Korteweg–de Vries (KdV) equation, while the second-order equation is solved numerically. The propagation velocity found for the solitary waves of the second-order equation coincides with a known expression, but it is obtained in a simpler way. By measuring the propagation velocity of solitary waves in the laboratory, we demonstrate that the second-order theory gives a considerably improved fit to experimental results.