Third-order Approximation Research Articles

Indirect nuclear spin-spin couplings with third-order contributions added to the SOPPA method.

In this article, a modification of the second-order polarization propagator approximation (SOPPA) method is introduced and illustrated for the calculation of the indirect nuclear spin-spin couplings. The standard SOPPA method, although cheaper in terms of computational cost, offers less accurate results than the ones obtained with coupled cluster methods. A new method, named SOPPA+A3-3, was therefore developed by adding the terms of the third-order A matrix that rely on the second-order double amplitudes. The performance of this third-order contribution was studied using the coupled cluster singles and doubles method as a reference, calculating the spin-spin couplings of molecules of diverse sizes and compositions, and comparing them to the SOPPA method. The results show that inclusion of this third-order contribution gives more accurate results than the standard SOPPA method with a level of accuracy close to that of the coupled cluster method with only a small increase in the computational cost of the response calculation that dominates the computational cost for small- to medium-sized molecules. The implementation of the first contributions to the third-order polarization propagator approximation in the Dalton program, thus, already shows a significant change in these molecular properties over those obtained with the standard SOPPA method.

CP3d: A comprehensive Euler-Lagrange solver for direct numerical simulation of particle-laden flows

We present a comprehensive Euler-Lagrange solver, CP3d, for the direct numerical simulation of particle-laden flows. The solver can handle one-way, two-way, interface-unresolved four-way and interface-resolved coupling regime. To make the solver versatile, different numerical approaches are available for fluid, particle as well as the coupling sub-solver. For the sake of computational accuracy and efficiency, a third-order exponential approximation method is proposed for Basset history force in interface-unresolved regime, and a new simplified lubrication force model based on the averaged integration is utilized in interface-resolved simulation to account for the short-range hydrodynamic force. The volume integration approach is also modified to adapt the staggered mesh configuration. The present solver is intensively validated against several benchmarks. A 2D pencil-like domain decomposition is implemented for parallel communication. The resulting solver is able to simulate large scale cases with billions of grid points and millions of moving particles in interface-resolved four-way regime, using only hundreds of computational cores. In addition, a nearly perfect linear strong scaling performance is achieved. In our test, CP3d is more than ten times faster than other similar solver reported in the literature. Program summaryProgram Title: Channel-Particle 3D, CP3d.CPC Library link to program files:https://doi.org/10.17632/7j6kcf8629.1.Developer's repository link:https://github.com/gongzheng-justin/CP3d.Code Ocean capsule:https://codeocean.com/capsule/0609843.Licensing provisions: MIT License.Programming language: Fortran 95, parallelized using MPI.External libraries: FFTW.Nature of problem:CP3d provides a comprehensive tool to perform the direct numerical simulation for particle-laden flows. One-way, two-way, interface-unresolved four-way and interface-resolved coupling regimes can be conducted.Solution method: the code employs second-/fourth-order finite-difference discretization for the fluid Navier-Stokes equation, and numerically solves the Maxey-Riley equation for the particle phase. In four-way regime, particle collisions are considered by discrete-element method. Immersed boundary method is utilized in the interface-resolved coupling regime.

Бикомпактные схемы для решения одногрупповой системы уравнений квазидиффузии совместно с уравнением энергии

High-order bicompact schemes are proposed for one of the stages of the HOLO algorithm for solving the radiative transport equation and implemented. The system of quasi-diffusion equations is solved together with the energy equation. The schemes have a fourth-order approximation in space on a two-point stencil, integration over time is carried out using the diagonal-implicit Runge–Kutta method of the third order of approximation, each stage of the method can be represented as an implicit Euler method. It is proposed to organize an iterative process to find a solution, since the coefficients of the system are non-linear functions of temperature. The schemes are used for a number of analytical tests, the convergence in time and space with the declared orders is shown.

Calculation of thermodynamic properties and transport coefficients of Ar-C-Si plasma

The compositions, thermodynamic properties and transport coefficients of the argon-carbon-silicon plasma at local thermodynamic equilibrium and local chemical equilibrium in temperatures range of 300－30000 K and pressure range of 0.1 to 10 atm and are different mixture ratios are investigated in this work. The condensed phases and Debye-Hückel corrections are both taken into account. The equilibrium component in gas phase is calculated by mass action law (Saha’s law and Gulberg-Waage’s law), Dalton’s partial pressure law, conservation of the elements and charge quasi-neutral equation, and at the same time the condensed species is calculated under the assumption of local phase equilibrium. Thermodynamic properties including density, enthalpy and specific heat are evaluated through a classical statistical mechanics approach. The transport coefficient calculations including viscosity, electrical conductivity, and thermal conductivity are carried out by using a third-order approximation (second-order for viscosity) of the Chapman-Enskog method. Collision integrals are obtained by using the relatively new data. The results show that the concentration and ratio of blend of C vapor and Si vapor can greatly affect the properties of the Ar plasma owing to the introduced C and Si vapor’ s own properties and their new reactions. While the pressure influences those properties through the shift of chemical equilibrium and the change of total number density. In addition, the introduction of condensed species leads the thermodynamic properties and transport coefficients of the lower temperature plasma to become almost the same as those of pure argon, and causes discontinuous points at phase-transition temperature. The final calculation results are in good agreement with those in the literature, and the slight difference in transport coefficient between them can be explained by the different selection of interaction potentials. The results are expected to provide reliable basic data for the numerical simulation of argon-carbon-silicon plasma.

Approximate analytical investigation of large-amplitude rolling motion of ships with the improved Galerkin method

In this study, the large-amplitude rolling motion of a ship was studied. First, the harmonic balance method was used to construct a reasonable initial approximate solution. Subsequently, the Newton linearization and Galerkin method were combined to obtain the analytical approximate solution of the ship's large-amplitude roll motion. By comparing with the numerical solution, the accuracy of the analytical approximation solution was verified. The results showed that the third-order analytical approximation had a higher accuracy for small and large amplitudes. Based on the analytical approximate solution, the influence of nonlinear quadratic damping and nonlinear restoring moment on the global stability of the ship under different amplitudes was analyzed. The theoretical results can guide the design and stability analysis of large rolling motions of ships.

A fully dynamic third-order EF-CIFF noise-shaping SAR ADC with NTF zeros optimization and passive integration

In this paper, a third-order noise-shaping successive approximation register (NS-SAR) analog-to-digital converter (ADC) is proposed which is simple and fully dynamic. A passive integrator is utilized in the feed-forward (FF) path of a second-order NS-SAR ADC with 2nd-order error-feedback (EF) filter to reduce the effect of inband quantization and comparator noise. The integrator implements a passive gain of 4x to make up its loss arising from the charge sharing between the integrator and digital-to-analog converter capacitors at the integration phase. Another advantage of the proposed structure is employing a simple low noise and low power one-input pair comparator unlike many published NS-SAR ADCs which use a comparator with multi-input pairs leading to enhanced comparator input-referred noise and power consumption. In this work, the zeros of the noise transfer function are optimized to enhance inband noise suppression. The proposed ADC is designed at circuit level using 0.18 μm TSMC CMOS technology with Cadence Virtuoso tool. The oversampling ratio, sampling frequency, and bandwidth are 8, 2.5 MS/s, and 156.25 kHz, respectively. According to the detailed post-layout simulation results, the achieved signal-to-noise and distortion ratio of the simulated ADC is 83.1 dB with 70.3 µW power consumption from a 1.1 V supply.

On solute dispersion in an oscillatory magneto-hydrodynamics porous medium flow under the effect of heterogeneous and bulk chemical reaction

It is well known to all of us that there is a shortage of pure drinking water across the globe. Different types of pollutants (metallic and nonmetallic) mix with the water, and they cause several diseases such as cholera, typhoid, and various kinds of skin diseases, and even it is found that these kinds of particles may cause skin cancer. In the current study, an analytical solution of a two-dimensional convection–diffusion equation is obtained using Mei's multi-scale homogenization technique to investigate the influences of homogeneous and heterogeneous reactions on dispersion phenomena of the solute in an oscillatory magneto-hydrodynamics porous medium flow. In the appearance of the applied transverse magnetic field and oscillatory pressure gradient, a mathematical model of magneto-hydrodynamics dispersion between two parallel plates is presented. The analytical expressions of Taylor dispersivity, longitudinal mean and real concentration distributions, transverse concentration distribution, and transverse uniformity rate of the concentration are obtained. Also, the effect of various flow parameters such as Péclet number, Hartmann number, Schmidt number, Darcy number, oscillatory Reynolds number, porous parameter, dispersion time, downstream and upstream locations, chemical heterogeneous boundary reaction, and bulk reaction is discussed. How the transport phenomena of the solute display different natures with the various ranges of Darcy and Hartmann numbers with the aid of homogeneous and heterogeneous boundary reactions are highlighted. To show the effect of the absorption parameters on the transport coefficient, the third-order approximation of concentration is performed. It is seen that the dispersion coefficient (DT1) corresponding to the purely time-dependent flow increases with the enhancement of the Darcy number (Da). Moreover, it is found that as the Hartmann number (M) enhances, the total dispersivity (DT) decreases. Also, the transverse concentration distribution becomes flat for larger values of the Hartmann number. It is noticed that when Da≥1, the transverse variation curve turns into a trimodal distribution from a bimodal. This model may be helpful for separating various metallic and nonmetallic particles from the water to reduce the water pollution.

Third-Order Effective Properties for Random-Packing Systems Using Statistical Micromechanics Based on a GPU Parallel Algorithm in Fast Computing n-Point Correlation Functions.

Estimating the effective properties of a particulate system is the most direct way to understand its macroscopic performance. In this work, we accurately evaluate the third-order approximations involving the three-point microstructural parameter , which can be calculated from a triple integral involving 1-, 2-, and 3-point correlation functions. A GPU-based parallel algorithm was developed for quickly computing the n-point correlation functions, and the results agree well with analytical solutions. The effective thermal conductivity and diffusion coefficient are calculated by the third-order approximates for the random-packing systems of a super-ellipsoid. By changing the parameters of the super-ellipsoid, the particle-shape effect can be predicted for both the thermal conductivity and diffusion coefficient.

Modeling of the vibration and stability of a dynamical system coupled with an energy harvesting device

This paper studies the vibrational motion of a dynamical system connected to an electromagnetic device, which is one of the energy harvesting (EH) devices that transform the vibrational motion into electric energy. This system has three degrees-of-freedom (DOF) and consists of two linked parts attached together; one is a nonlinear Duffing oscillator, and the other is a nonlinear damping spring pendulum. The regulating equations of motion (EOM) are achieved utilizing Lagrange’s equations and solved analytically applying the approach of multiple scales (AMS) till the third order of approximation. The accuracy of the attained solutions has been examined by comparing them with the numerical ones of the EOM. The time histories of the solutions and the nonlinear stability analysis of the modulation equations are represented graphically in various plots. The Poincaré maps and phase portraits diagrams displayed the stable behavior of the studied dynamical system. In addition, the different ranges of the stabilities are examined and discussed. In the electromagnetic device, the output power and current time series are depicted as a function of different values of the damping coefficients, excitation amplitudes, and load resistance. It is noted that the output current and power are dropped when the damping coefficient is raised. On the other hand, the increment of the excitation has a positive effect on the electrical generation and produces increment of the output power and current. Furthermore, the output power grows when the total resistance increases to accommodate the applied load. The EH device generates high output current and power at low-frequency values. The significance of this work is limited to the numerous uses of its outcomes in everyday life, such as powering medical devices, serving as a power supply for sensors, and serving as a backup energy source for some electronic devices.

APPROXIMATE SOLUTIONS OF WEAKLY NONLINEAR DIFFERENTIAL EQUATIONS

In this work, we study the most useful approximation methods for proving solutions to analytic approximation for solving a weak second-order nonlinear differential equation in a power series with small parameters. We prove the second-order periodic approximation solution and also the best third-order approximation of the weak nonlinear differential equation.

Quasinormal modes of the Hayward black hole surrounded by quintessence: Scalar, electromagnetic and gravitational perturbations

In this paper, we study the quasinormal modes for scalar, electromagnetic, and gravitational axial perturbations in the Hayward regular black hole surrounded by quintessence (HBH-[Formula: see text]). Using the third-order WKB approximation, we can determine the dependence of the quasinormal modes on the parameters of the regular black hole and the parameters on the test fields. We also determine the greybody factor, giving transmission and reflection coefficients of the scattered wave through the effective potentials in the WKB approximation using numerical analysis.

TVD-модификация метода Годунова 3-го порядка

A study of the monotonicity properties of a new modification of the Godunov method, which has a third order of approximation in space and time, is presented. The difference scheme of the method uses a simultaneous discretization of conservation laws in space and time without of Runge — Kutta stages. This method is a development of the second order Godunov method through the connection of two additional fluxes correction procedures. The first procedure increases the order of approximation for linear systems of equations. The second procedure uses the finite differences of the Jacobi matrix of the system of equations and eliminates the second-order error that occurs due to the nonlinearity of the equations. The TVD property of the difference scheme is strictly proved in relation to the linear scalar transfer equation when using an generalized adaptive limiter of central differences. New modifications of limiters are proposed that significantly improve the accuracy of the solution in the vicinity of discontinuities and local extremes. New limiters are obtained from the known ones using a simple function of local smooth deformation. An economical version of the third-order Godunov method for gas dynamics equations with quadratic reconstruction of parameters in primary variables is presented. The use of primary variables for reconstruction significantly reduces the FPU time during calculations. The advantages of the third-order method in terms of the accuracy of the solution in the vicinity of contact discontinuities and the boundaries of the rarefaction wave are demonstrated on standard tests. The proposed approach to constructing third-order difference schemes can be used for inhomogeneous and two-dimensional hyperbolic systems of nonlinear equations.

Analyzing the Stability for the Motion of an Unstretched Double Pendulum near Resonance

This work looks at the nonlinear dynamical motion of an unstretched two degrees of freedom double pendulum in which its pivot point follows an elliptic route with steady angular velocity. These pendulums have different lengths and are attached with different masses. Lagrange’s equations are employed to derive the governing kinematic system of motion. The multiple scales technique is utilized to find the desired approximate solutions up to the third order of approximation. Resonance cases have been classified, and modulation equations are formulated. Solvability requirements for the steady-state solutions are specified. The obtained solutions and resonance curves are represented graphically. The nonlinear stability approach is used to check the impact of the various parameters on the dynamical motion. The comparison between the attained analytic solutions and the numerical ones reveals a high degree of consistency between them and reflects an excellent accuracy of the used approach. The importance of the mentioned model points to its applications in a wide range of fields such as ships motion, swaying buildings, transportation devices and rotor dynamics.

Local discontinuous Galerkin method for the numerical solution of fractional compartmental model with application in pharmacokinetics

This paper provides a numerical solution for the fractional multi-compartmental models which are applied in pharmacokinetics. We implement the local discontinuous Galerkin method for these fractional models with the upwind numerical fluxes. To obtain high-order results with adequate accuracy, the third-order approximation polynomials are used. Finally, to validate the scheme, the results are compared with the solutions of a semi-analytical method.

Multireference Algebraic Diagrammatic Construction Theory for Excited States: Extended Second-Order Implementation and Benchmark.

We present an implementation and benchmark of new approximations in multireference algebraic diagrammatic construction theory for simulations of neutral electronic excitations and UV/vis spectra of strongly correlated molecular systems (MR-ADC). Following our work on the first-order MR-ADC approximation [J. Chem. Phys. 2018, 149, 204113], we report the strict and extended second-order MR-ADC methods (MR-ADC(2) and MR-ADC(2)-X) that combine the description of static and dynamic electron correlation in the ground and excited electronic states without relying on state-averaged reference wave functions. We present an extensive benchmark of the new MR-ADC methods for excited states in several small molecules, including the carbon dimer, ethylene, and butadiene. Our results demonstrate that, for weakly correlated electronic states, the MR-ADC(2) and MR-ADC(2)-X methods outperform the third-order single-reference ADC approximation and are competitive with the results from equation-of-motion coupled cluster theory. For states with multireference character, the performance of the MR-ADC methods is similar to that of an N-electron valence perturbation theory. In contrast to conventional multireference perturbation theories, the MR-ADC methods have many attractive features, such as a straightforward and efficient calculation of excited-state properties and a direct access to excitations outside of the frontier (active) orbitals.

Image Compression and Enlargement Algorithms

In some cases, there are problems associated with the compression and enlargement of images. The use of splines is quite effective in some cases. In this paper, a new image compression algorithm is presented. The features of increasing the size of an image when using local polynomial or non-polynomial splines are considered. The main method for enlarging an image is based on the use of splines of the second and third order of approximation. Polynomial and trigonometric splines are considered. To speed up the process of enlarging the image, we used the parallelization techniques

A new pendulum motion with a suspended point near infinity

In this paper, a pendulum model is represented by a mechanical system that consists of a simple pendulum suspended on a spring, which is permitted oscillations in a plane. The point of suspension moves in a circular path of the radius (a) which is sufficiently large. There are two degrees of freedom for describing the motion named; the angular displacement of the pendulum and the extension of the spring. The equations of motion in terms of the generalized coordinates varphi and xi are obtained using Lagrange’s equation. The approximated solutions of these equations are achieved up to the third order of approximation in terms of a large parameter varepsilon will be defined instead of a small one in previous studies. The influences of parameters of the system on the motion are obtained using a computerized program. The computerized studies obtained show the accuracy of the used methods through graphical representations.

Temperature dependence of thermal expansion of solids: A new approach in the terms of the Lambert function

The new equation for the temperature dependence of the thermal expansion of solids is explicitly presented in terms of the Lambert function. The resulting expression is suitable for a wide class of solids in the entire temperature range of the solid phase, in which, upon heating, there are no other phenomena causing changes in atomic volume, such as phase or glass transitions and the formation of vacancies. At temperatures well below the Debye temperature, the new equation provides a linear correlation between thermal expansivity and specific heat in accordance with Grüneisen's law. Whereas at intermediate temperatures not exceeding half the melting point, the [1/1] and [1/2] Padé approximants of the present equation reduce to the Grüneisen expressions in the second and third order of approximation. Unlike the previous ones, the new equation predicts a sharp increase in expansivity at high temperatures in accordance with experiment.

A fourth-order approximation Rayleigh-Plesset equation written in volume variation for an adiabatic-gas bubble in an ultrasonic field: Derivation and numerical solution

The derivation of a nonlinear ordinary differential equation for modeling the nonlinear oscillations of a gas bubble placed in an ultrasonic field is performed in terms of bubble-volume variations up to the fourth-order approximation. The equation, written within the Rayleigh-Plesset framework, is solved through numerical approximations. Results from simulations are compared to data obtained from the classic second-order approximation equation derived in the 1960–70’s, usually used in this framework, and from the third-order approximation equation derived in the 1990’s. This comparison shows that the fourth-order approximation allows us to observe the nonlinear behavior of the bubble at high finite amplitude, which differs from the other approximations when the nonlinearity of the phenomenon is higher, i.e., when the driving acoustic frequency is close to the bubble resonance.

Weakly nonlinear evolution of pressure waves in a polydisperse bubbly liquid

Weakly nonlinear propagation of plane pressure waves in initially quiescent liquids containing spherical bubbles with a small initial polydispersity of the bubble radius and bubble number density is theoretically studied to clarify the effect of the polydispersity on wave propagation. The wavelength is comparable with the bubble radius and the incident frequency of waves is with an eigenfrequency of the bubble. We consider two different cases of the size of the polydispersity (i.e., small and large polydispersities) and assume that the polydispersity appears at a field far from a sound source. The polydispersity is formulated into a set of perturbation expansions. Bubble oscillations are spherically symmetric, and bubbles do not coalesce, break up, appear, and disappear. The basic set is composed of the conservation laws of mass and momentum for gas and liquid phases based on a two-fluid model, equation of motion for bubble wall, and so on. From the perturbation analysis up to the third order of approximation, two cases of nonlinear wave equation for a long-range propagation of waves were derived. We concluded that the polydispersity affects the advection of waves and is expressed as a variable coefficient.