Third-order Approximation Research Articles

Higher-order approximations for space-fractional diffusion equation

Second-order and third-order finite difference approximations for fractional derivatives are derived from a recently proposed unified explicit form. The Crank-Nicholson schemes based on these approximations are applied to discretize the space-fractional diffusion equation. We theoretically analyse the convergence and stability of the Crank-Nicholson schemes, proving that they are unconditionally stable. These schemes exhibit unconditional stability and convergence for fractional derivatives of order in the range . Numerical examples further confirm the convergence order and unconditional stability of the approximations, demonstrating their effectiveness in practice.

The Hermite-Birkhoff Problem and Local Spline Approximation

This paper discusses the use of local spline approximations to solve the Hermite-Birkhoff problem. The solution to a specific problem using polynomial and non-polynomial local splines of the third order of approximation is considered. Here we discuss the case when the values of the function 𝑢(𝑥) and its derivative 𝑢’(𝑥) are given at the nodes of the grid in an alternative way: … , 𝑢(𝑥𝑗), 𝑢′(𝑥𝑗+1), 𝑢(𝑥𝑗+2), … . Note that when using polynomial and non-polynomial spline approximations, it is possible to obtain acceptable solutions in several interesting cases that are impossible when we use the classical approach. In the case of using local basis splines, many previously unsolvable problems turn out to be solvable. The results of the numerical experiments are presented.

Calculation of thermodynamic properties and transport coefficients of Ar/N2–H2–Si plasma

Abstract The composition, thermodynamic properties and transport coefficients of Ar – H 2 – Si and N 2 – H 2 – Si plasma within a temperature range of 300–30 000 K and pressure range of 0.1–10 atm are calculated under the assumptions of local thermal equilibrium (LTE) and local chemical equilibrium (LCE). Taking Debye–Hückel corrections into account, the chemical equilibrium composition and thermodynamic properties of these two plasma systems are derived using the mass action law and classical statistical thermodynamics respectively. The transport coefficients, including viscosity, conductivity, and thermal conductivity, are calculated using the Chapman–Enskog (C–E) method extended to a third-order approximation (second-order for viscosity and heavy particle translational thermal conductivity). Some of the results have been compared with those of other researchers, and there is a good level of agreement. The slight difference arises from the selection of interaction potential. The final calculation reveals that the introduction of silicon vapor significantly alters the thermodynamic properties and transport coefficients of Ar / N 2 – H 2 – Si plasma, even at a small concentration of silicon vapor (1%), the effect on the electrical conductivity cannot be ignored. Furthermore, our calculation results provide the fundamental data for numerical simulations of magnetohydrodynamics (MHD) for the synthesis of silicon nanoparticles and silicon composites.

Статистический анализ изменения температуры воздуха по данным многолетних наблюдений Метеорологической обсерватории имени В. А. Михельсона РГАУ – МСХА

Introduction. The purpose of the study is to assess the dynamics of changes in air temperature indicators and its impact on agronomic systems, based on long-term observations of the V. A. Mikhelson Meteorological Observatory of RSAU – MTAA. Materials and Methods. Using the Mathcad program, graphs of changes in average annual temperatures were constructed using a third-order approximation polynomial, graphs of the theoretical normal density distribution and normal temperature distribution function, as well as graphs of intra-annual temperature distribution for 30 years. Research results and Discussion. The values of the relative error of average annual temperatures were calculated, the normal distribution of average annual temperatures was checked using the Pearson criterion, after which the probability of deviation of the empirical hours falling into intervals from the theoretical hours was found. Conclusion. As a result of the research, a significant increase in average annual air temperature was established against the background of a general reduction in the duration of the cold period. Increase in temperature over the period 1961–1990 amounted to 0.9 °C, which is significantly higher than the average values for the globe. The results of the study reflect the general trend of warming in the context of global climate change and can be used to improve the forecasting of weather conditions, as well as to optimize technological processes in crop production.

Anticipating Pressure Changes in Halides under Compression

A new equation of state (NEOS) for Halides has been developed using the theory of lattice potential and the concept of volume dependence of the short-range force constant. The derivation of this equation of state involved the use of the third-order approximation of the lattice potential. A comparative analysis was conducted between the isothermal equations of state, including Vinet EOS, Murnaghan EOS, Holzapfel EOS, Born-Mie EOS, Birch-Murnaghan EOS, and the newly derived NEOS. The NEOS was used to analyze the compression behavior of Halides, and it was found that Vinet EOS and NEOS agreed with the experimental data for Halides up to high compression. However, Murnaghan EOS, Born-Mie EOS, Holzapfel EOS, and Birch-Murnaghan EOS are usually less sensitive to calculating pressure at high compression. It was also observed that for some Halides, such as NaBr and NaI, Vinet EOS could not produce results consistent with experimental findings. In contrast, NEOS consistently produced results that matched the experimental findings for all Halides samples, unequivocally demonstrating its reliability and accuracy.

Primary Resonance of Nonlinear Spinning Timoshenko Shaft Based on a Novel Third-order Approximation Model Derived from Geometrically Exact Nonlinear Model

Primary Resonance of Nonlinear Spinning Timoshenko Shaft Based on a Novel Third-order Approximation Model Derived from Geometrically Exact Nonlinear Model

Investigation of an optimization method for time constants to calibrate reaction calorimeters

In exothermic reactions, scale-up sometimes endangers the chemical process because of the risk of runaway reactions. Reaction calorimeters are used in a laboratory to design a safe chemical process by collecting thermal and kinetic parameters for scale-up. Because scale-up can magnify the errors obtained in laboratory experiments, the cooling capacity can be improperly calculated based on the underestimated heat generation due to thermal lag. Therefore, reaction calorimeters are required to provide accurate heat flow despite thermal delays inherent in the reaction calorimeters.This study validates the time-constant search method for a reaction calorimeter that employs a function of smoothness S(τ) in the presence of noise. The degree of noise was determined by measuring the actual value using a reaction calorimeter. The simulation conducted using spreadsheets demonstrated that even in the presence of noise, it is possible to achieve high accuracy in the search for time-constants when there is a small thermal lag and a large intensity of calibration heat. Moreover, the use of a third-order polynomial approximation to calculate the derivative value proved to be suitable for determining the time constant accurately.

Effect of thermal diffusivity on Rayleigh waves in human tissue

This work examines the nature of human tissue under the context of dual phase lag thermoelastic diffusion theory using the principle of propagation of Rayleigh wave along the stress free, thermally insulated/isothermal, and impermeable/isoconcentrated conditions. The governing equations for the tissue are formed by considering the third-order approximation of Taylor’s expansion for the modified Fourier’s law of heat conduction and modified Fick’s law of mass diffusion in terms of phase lag parameters. The irrational secular equations of Rayleigh waves are derived, which are then rationalized to extract their complex roots. The phase speeds and the attenuation coefficients are evaluated from these complex roots. The variation of these phase speeds and attenuation coefficients are plotted graphically against angular frequency for two significant modes. The behavioral patterns of displacement components, temperature, and concentration fields are computed and are expressed graphically. The paths of the particle motion are represented graphically. Some results of published articles are recovered from the present formulation to validate our results.

Transport cross sections and collision integrals for N(4S)–O(3P) and N(4S)–O(1D) interactions in high-temperature air plasmas

Collisions between nitrogen (N) and oxygen (O) play a crucial role in determining transport coefficients in high-temperature atmospheres of Earth and planetary. In this study, the momentum transfer, viscosity, third-moment, and fourth-moment transport cross sections for the N(4S)–O(3P) and N(4S)–O(1D) interactions are reported in the collision energy range of 10−6–10 Hartree based on the classical and semiclassical methods. The new and accurate potential energy curves for N–O interactions, which are used to provide the input for calculations of the cross sections, are calculated based on the state-of-the-art ab initio method. The classical and semiclassical collision integrals are provided at 300–50 000 K, and the results support the calculation of transport coefficients in a third-order approximation. In particular, the collision data for the N(4S)–O(1D) interaction based on ab initio points are reported for the first time. The calculated transport cross sections and collision integrals are helpful for studies of modeling the high-temperature air plasmas.

The effects of nonlinear approximation to mass operator on the wave propagation in polycrystalline materials

Existing theoretical models utilize the first or third-order smoothing approximations (FOSA and TOSA) to predict attenuation and wavespeed in polycrystalline materials. The FOSA-based model involves a Born approximation, limiting its application to the stochastic (low-frequency) scattering regime. This restriction is a significant factor contributing to its disagreement with FE-based models. The model based on TOSA was subsequently developed, incorporating two additional multiple scattering terms, and increasing the degree of heterogeneity. To extend into the high-frequency regime, the TOSA-based model was solved using numerical integrations. This presentation highlights the analytical solution of the Dyson equation based on the non-linear approximation, wherein the mass operator expression incorporates the mean dyadic Green’s function. Evaluating this equation with consideration of higher-order correlations addresses potential truncation errors associated with FOSA and TOSA approximations and encompasses the full range of frequencies from Rayleigh to the geometric asymptote. This presentation will also demonstrate a comparison of the results obtained from the nonlinear approximation with those from FOSA and TOSA across a range of polycrystalline metals.

Implications of a Simpson–Visser solution in Verlinde’s framework

This study focuses on investigating a regular black hole within the framework of Verlinde’s emergent gravity. In particular, we explore the main aspects of the modified Simpson–Visser solution. Our analysis reveals the presence of a unique physical event horizon under certain conditions. Moreover, we study the thermodynamic properties, including the Hawking temperature, the entropy, and the heat capacity. Based on these quantities, our results indicate several phase transitions. Geodesic trajectories for photon-like particles, encompassing photon spheres and the formation of black hole shadows, are also calculated to comprehend the behavior of light in the vicinity of the black hole. Additionally, we also provide the calculation of the time delay and the deflection angle. Corroborating our results, we include an additional application in the context of high-energy astrophysical phenomena: neutrino energy deposition. Finally, we investigate the quasinormal modes using third-order WKB approximation.

Реализация бикомпактной схемы для HOLO алгоритма решения задач переноса излучения в среде

Bicompact schemes for the HOLO algorithm for solving the transport equation were applied to solve model problems of radiation transport in a medium (the first and the second Fleck problems). The main idea of HOLO algorithms is to accelerate the convergence of iterative processes due to the joint solving of high order (HO) and low order (LO) kinetic equations. The schemes are constructed by the method of lines on a minimal two-point stencil and have the fourth order of approximation in space. The Runge-Kutta method of the third order of approximation was used for integration over time. Verification and validation of the schemes were carried out. Solutions of the first and the second Fleck problems by the proposed method coincided well with solutions obtained by other methods. A modification of the second Fleck problem was proposed to investigate the dependence of the velocity of the thermal wave front on the absorption coefficient in the optically thick region.

Монотонизация модифицированной схемы с эрмитовой интерполяцией для численного решения неоднородного уравнения переноса с поглощением

A hybrid scheme for the numerical solving of a transport equation is implemented. The high order scheme is a modified CIP (Cubic Interpolation Polynomial) scheme with Hermitian interpolation. The third order of approximation of the CIP scheme is achieved by including in the list of unknowns not only the nodal values of the function, but also the nodal values of its derivatives. In the modification under consideration the Euler–Maclaurin formula is used to calculate derivatives. The low order scheme is the characteristic scheme of the first order of approximation. Local, layerwise and global monotonization are considered, the hybridization is performed, respectively, after a cell, a time layer or the whole grid processing. It is shown that the best results are obtained by a scheme with local monotonization. The convergence orders of the hybrid scheme on tests of different smoothness of exact solution do not differ significantly from the convergence orders of the CIP scheme. It is proposed to calculate an integral along a characteristic using the Simpson formula for the Stieltjes integral in the case of the large optical thickness; it provides a significant reduction in the numerical solution errors.

Higher-order scattering of high-frequency ultrasound in strongly heterogeneous microstructures

Ultrasonic waves get scattered from grain boundaries while propagating through polycrystals, thereby experiencing attenuation and a change in wavespeed. These grain boundaries act as the sole scattering sites for untextured aggregates of randomly oriented crystallites. In addition, porosity, texture, defects, precipitates, or inclusions also serve as scatterers when present in a microstructure. Existing analytical models rely on the first-order smoothing approximation (FOSA)-based homogenization to the mass operator series in the governing Dyson equation. They can accurately predict ultrasonic dispersion in some weakly inhomogeneous metals like Aluminum. However, while comparing against more realistic Finite Element (FE) predictions, serious discrepancies, as high as 70%, are found in the attenuation estimates for metals, like Lithium, that possess strong elastic fluctuations in their microstructures. Thus, the current model analytically investigates higher-order scattering effects corresponding to the third-order smoothing approximation (TOSA) for the first time. Current results reveal no higher-order scattering effects for Aluminum at any frequencies. However, the longitudinal attenuation estimates for Lithium are found to improve by 1.27% and 0.37% in the “stochastic” (high-frequency) and “Rayleigh” (low-frequency) regimes, respectively. The presentation will highlight the effects of including TOSA on analytical estimates of high-frequencyultrasonic dispersion in different strongly heterogeneous microstructures.

Hierarchical approximations to the nucleation work in the entire range of metastability.

The work W to form a nucleus (also known as the critical nucleus) is a key quantity in the description of nucleation phenomena because of its exponentially strong effect on the nucleation rate. The present study provides a general approximate expression for W, which comprises a hierarchy of approximations to the dependence of W on the experimentally controlled overpressure Δp of a nucleating multicomponent phase. This general expression is used to derive explicit formulas for the lowest-order members of the W(Δp) hierarchy as well as for the respective lowest-order approximations to the Δp dependences of the nucleus surface tension, the nucleus radius, the Gibbs-Tolman length, and the stationary nucleation rate. The second-order and the third-order approximations to the W(Δp) dependence are confronted with available W(Δp) data, and the latter is found to agree very well with the data. The results obtained are applicable to homogeneous single-component or multicomponent nucleation from the binodal to the spinodal of the old phase, i.e., in the entire range of the old-phase metastability.

Revisiting Universal Variables for Robust, Analytical Orbit Propagation Under the Vinti Potential

To meet the growing complexity and demands of modern spacecraft missions, analytical solutions to initial value problems see continued use, typically supporting global searches of large trajectory design spaces. These efforts often employ universal two-body orbit propagators for their recognized speed and robustness, but many applications, like active space debris removal, would benefit from a comparable propagator with greater accuracy. Vinti propagators, which consider planetary oblateness, may serve this purpose, but existing Vinti solutions possess computational difficulties in certain orbital regimes. To mitigate these deficiencies, the present study develops and validates an analytical, third-order universal Vinti propagator free of computational difficulties by leveraging standard, oblate spheroidal (OS) universal variables and OS equinoctial orbital elements. Accuracy of the third-order approximation is assessed for multiple examples across an array of orbital regimes. Computational runtime is also evaluated, and performance is directly compared to the benchmark Vinti6 algorithm. On average, the Vinti propagator implemented in this work is only slower than a typical universal Kepler propagator by a factor of 4.0 and slower than Vinti6 by a factor of 1.8, but with greater robustness than the benchmark. The new form of the equations of motion also has favorable implications for the associated boundary value problem.

A new constitutive model and hot processing map of 5A06 aluminum alloy based on high-temperature rheological behavior and higher-order gradients

To develop a high-precision constitutive model and hot processing map for 5A06 aluminum alloy, the hot deformation behavior of the alloy was investigated through isothermal compression experiments and microstructure analysis experiments, which were conducted at temperatures ranging from 573 K to 773 K and at strain rates ranging from 0.01 to 10 s−1. Firstly, a new constitutive model, which is based on the partial derivatives of the experimental flow data and does not significantly increase the material parameters, was proposed. Secondly, a comparison and analysis of the predictive accuracies of both the new model and the classic model were conducted. Finally, the new constitutive model was used to construct the hot processing maps of the 5A06 aluminum alloy, and the optimal hot processing parameters were validated through metallographic experiments. The results show that a high-precision constitutive model can be constructed with a small number of material parameters by considering the second-order approximation between logarithmic stress and temperature, as well as the third-order approximation between logarithmic stress and logarithmic strain rate. The mean square errors of the new model were significantly lower than those of the Arrhenius model and Hense-Spittle model for all strain rates and temperatures. There were significant differences in the prediction accuracy of the AH and HS models at different temperatures and strain rates, indicating their accuracy was dependent on temperature and strain rate. The coefficients of the new model were significantly higher than those of the AH and HS models. Moreover, the proximity of predicted values to experimental values was ranked in descending order as follows: new model, AH model, and HS model. The energy dissipation rate is relatively high when the strain rate is lower than 10 s-1 and the temperature is higher than 623 K. The instability factor is larger than zero for all conditions except when the strain rate is equal to 10 s−1. The optimal hot working temperature range for 5A06 aluminum alloy is 623–700 K, and the strain rate should be in the range of 0.01 s−1 to 1 s−1. The microstructure analysis matches the predicted results from hot processing maps, which can guide the hot working process of 5A06 aluminum alloy.

Aerodynamic analysis of a downwind offshore floating wind turbine with rotor uptilt angles in platform pitching motion

In this study, the effect of the rotor uptilt angle on the aerodynamic performance of a downwind floating wind turbine (FWT) is investigated using the free vortex wake (FVW) method. The Weissinger-L blade lifting model and a three-step, third-order predictor-corrector difference approximation of wake solution are applied in the FVW model. To consider the platform pitching angle and angular velocity, the blade inflow conditions and the positions of the first three nodes of vortex filaments are modified in the implemented model. The full-scale FWT simulation results, focusing on platform pitching motion, are validated by comparing them with results obtained using other established numerical methods, ensuring the accuracy and reliability of the presented model. Furthermore, the simulation of the wake field is also validated by comparison between calculation and experiment. In this setting, the comparison of the aerodynamic performance between upwind and downwind FWTs shows that the downwind FWT with a rotor uptilt angle has slightly higher power and thrust. The results indicate that the rotor uptilt angle has a significant effect on blade aerodynamic loads, with the highest impact observed at maximum pitching velocity and angle. In this context, at the position with maximum pitching velocity, a larger rotor uptilt angle results in a larger angle of attack and increased aerodynamic load, while at the maximum pitching angle position, a larger rotor uptilt angle leads to a smaller angle of attack and decreased aerodynamic load.

Reinforcement Learning in a New Keynesian Model

We construct a New Keynesian (NK) behavioural macroeconomic model with bounded-rationality (BR) and heterogeneous agents. We solve and simulate the model using a third-order approximation for a given policy and evaluate its properties using this solution. The model is inhabited by fully rational (RE) and BR agents. The latter are anticipated utility learners, given their beliefs of aggregate states, and they use simple heuristic rules to forecast aggregate variables exogenous to their micro-environment. In the most general form of the model, RE and BR agents learn from their forecasting errors by observing and comparing them with each other, making the composition of the two types endogenous. This reinforcement learning is then at the core of the heterogeneous expectations model and leads to the striking result that increasing the volatility of exogenous shocks, by assisting the learning process, increases the proportion of RE agents and is welfare-increasing.

Power spectra of slow-roll inflation in the consistent D → 4 Einstein-Gauss-Bonnet gravity

The slow-roll inflation which took place at extremely high energy regimes is in general believed to be sensitive to the high-order curvature corrections to the classical general relativity (GR). In this paper, we study the effects of the high-order curvature term, the Gauss-Bonnet (GB) term, on the primordial scalar and tensor spectra of the slow-roll inflation in the consistent D → 4 Einstein Gauss-Bonnet (4EGB) gravity. The GB term is incorporated into gravitational dynamics via the re-scaling of the GB coupling constant α → α/(D-4) in the limit D → 4. For our purpose, we calculate explicitly the primordial scalar and tensor power spectra with GB corrections accurate to the next-to-leading order in the slow-roll approximation in the slow-roll inflation by using the third-order uniform asymptotic approximation method. The corresponding spectral indices and their runnings of the spectral indices for both the scalar and tensor perturbations as well as the ratio between the scalar and tensor spectra are also calculated up to the next-to-leading order in the slow-roll expansions. These results represent the most accurate results obtained so far in the literature. In addition, by studying the theoretical predictions of the scalar spectral index and the tensor-to-scalar ratio with Planck 2018 constraint in a model with power-law potential, we show that the second-order corrections are important in future measurements.