Higher-order Approximations Research Articles

A family of high order approximations of Ramanujan type for perimeter of an ellipse

Inspired by Ramanujan’s unusual approximations for the perimeter of an ellipse with semiaxis 1 and $$r\in \left( 0,1\right) $$ , a family of approximations of Ramanujan type is constructed. We find the sharp lower and upper approximations have very high accuracy, and better than some known ones.

Higher-order stability analysis of a rotating BDFG tapered beam with time-varying velocity

In this paper, dynamic stability analysis is carried out for a rotating tapered cantilever beam made of bi-directional functionally graded (BDFG) materials with time-dependent rotating velocity. Rayleigh-Ritz method is employed to obtain the eigenfrequencies and mode shapes of the beam. Hamilton’s principle and the Galerkin method are adopted to establish the equation of motion with periodic coefficients. Dynamic instability problem due to the periodic rotating velocity is solved by Bolotin’s method with higher-order approximation. Both the sub-harmonic and harmonic parametric instability regions are investigated. Floquet theory is applied to verify the dynamic instability boundaries. The results indicate that the typical first-order approximate Bolotin’s method cannot meet the accuracy requirements of dynamic instability analysis for rotating BDFG tapered beam, and a second-order approximation is needed to improve computation accuracy. The effects of mean rotating velocity, hub radius, dynamic amplitude factor, material FG indexes and taper ratio on the natural frequencies and dynamic instability characteristics of the rotating BDFG tapered beam are discussed under different parameters.

A Unifying Framework for Perturbative Exponential Factorizations

We propose a framework where Fer and Wilcox expansions for the solution of differential equations are derived from two particular choices for the initial transformation that seeds the product expansion. In this scheme, intermediate expansions can also be envisaged. Recurrence formulas are developed. A new lower bound for the convergence of the Wilcox expansion is provided, as well as some applications of the results. In particular, two examples are worked out up to a high order of approximation to illustrate the behavior of the Wilcox expansion.

Efficient projection kernels for discontinuous Galerkin simulations of disperse multiphase flows on arbitrary curved elements

In this work, we develop projection kernels for Euler-Lagrange point-particle simulations of disperse multiphase flows on arbitrary curved elements. These kernels are employed in a high-order discontinuous Galerkin framework for projecting the action of the particles to the Eulerian mesh. Instead of commonly used isotropic kernels, such as a Gaussian-type kernel, we construct an anisotropic polynomial-based smoothing function that preserves the compactness of the discontinuous Galerkin method on high-aspect-ratio elements and maintains an acceptable computational cost. At the same time, it mitigates inaccuracies associated with larger particles and numerical instabilities arising from the Dirac delta function and low-order kernels. Specifically, the geometric mapping of a physical element to the reference element is exploited to construct a kernel that is elliptical in 2D and ellipsoidal in 3D. We also develop a strategy to conserve interphase transfer near boundaries, particularly on curved elements. This strategy employs a high-order polynomial approximation to appropriately rescale the interphase source terms in an efficient manner. The compatibility of the proposed methodology with different types of meshes is investigated. We then apply it to a number of particle-laden flow configurations, including supersonic dusty flow over a flat plate, moving shocks interacting with clouds of particles, and hypersonic dusty flows over blunt bodies.

High-Accuracy Impedance Read-out Circuit for BIA-type Biomedical Sensors

An impedance read-out circuit with high accuracy implemented using a TSMC 0.18 um mixed-signal CMOS process for BIA-type (bio-electrical impedance analysis) sensors to detect biomarkers is proposed in this investigation. In addition to proposing the impedance estimation circuit, the associated analytic proof is also disclosed. Using a high-order approximation model and high noise rejection capability by the addition of OPAs (operational amplifiers), the proposed system demonstrates a solution with high accuracy in terms of the maximum impedance error ( $$\mathrm<$$ 6 %) and the least phase error ( $$\mathrm < $$ 3 $${^\circ }$$ ) in a frequency range from 1 kHz to 2 MHz.

Distributed Car-Following Control for Intelligent Connected Vehicle Using Improved Super-Twisting Compensator Subject to Sudden Velocity Changes of Leading Vehicle

The optimal velocity-based model has been successfully applied to distributed car-following systems. However, the car-following performance is inevitably affected by a series of disturbances, particularly, sudden velocity changes of leading vehicle. To improve accuracy and response rate of car-following control in the presence of such disturbances, an improved super-twisting compensator (ISTC) is proposed and a composite controller is designed by combining ISTC with a finite- time controller. A second-order nominal system is constructed by using a virtual measurement signal along with its integration to facilitate the design of ISTC. By introducing the feedback of high-order estimation error, the accuracy and response rate of ISTC are increased significantly as compared with the conventional one under same gains. Such improvement further enhances the disturbance rejection ability of the composite controller. Both Lyapunov approach and numerical simulations are carried out to verify the effectiveness of the proposed method.

Schauder estimates on smooth and singular spaces

In this paper, we present a proof of Schauder estimate on Euclidean space and use it to generalize Donaldson’s Schauder estimate on space with conical singularities in the following two directions. The first is that we allow the total cone angle to be larger than 2\(\pi \) and the second is that we discuss higher-order estimates.

Higher-Order Weak Schemes for the Heston Stochastic Volatility Model by Extrapolation

We consider a time-discrete scheme for the Heston stochastic volatility model, which employs the stochastic trapezoidal rule to discretize the logarithmic asset process, provided that the variance process is simulated exactly. We prove that, with respect to any polynomial function of the log-asset process, the weak error can be expanded to arbitrarily high powers of step size, which allows us to construct higher-order weak approximations by extrapolation. The result applies for the full parameter regime.

The design of new high-order group iterative method in the solution of two-dimensional fractional cable equation

In this article, a new high-order explicit group iterative scheme is developed for the solution of the two-dimensional time fractional cable equation. The proposed scheme is derived from the Crank-Nicolson (C-N) high-order compact finite difference method, where the Caputo discretization and C-N high-order approximations are used for the time fractional and space derivative respectively. The stability and convergence of the proposed scheme are also established. Finally, two numerical examples are presented to show the accuracy and efficiency of the scheme.

Atom-dimer and dimer-dimer scatterings in a spin-orbit-coupled Fermi gas

Using the diagrammatic approach, here we study how spin-orbit coupling (SOC) affects the fermion-dimer and dimer-dimer scattering lengths in the Born approximation, and benchmark their accuracy with the higher-order approximations. We consider both isotropic and Rashba couplings in three dimensions, and show that the Born approximation gives accurate results in the $1/(m \alpha a_s) \ll -1$ limit, where $m$ is the mass of the fermions, $\alpha$ is the strength of the SOC, and $a_s$ is the $s$-wave scattering length between fermions. This is because the higher-loop contributions form a perturbative series in the $1/(m \alpha a_s) < 0$ region that is controlled by the smallness of the residue $Z$ of the dimer propagator. In sharp contrast, since $Z$ grows with the square-root of the binding energy of the dimer in the $1/(m \alpha a_s) > 0$ region, all of the higher-loop contributions are of similar order.

High order pressure-based semi-implicit IMEX schemes for the 3D Navier-Stokes equations at all Mach numbers

This article aims at developing a high order pressure-based solver for the solution of the 3D compressible Navier-Stokes system at all Mach numbers. We propose a cell-centered discretization of the governing equations that splits the fluxes into a fast and a slow scale part, that are treated implicitly and explicitly, respectively. A novel semi-implicit discretization is proposed for the kinetic energy as well as the enthalpy fluxes in the energy equation, hence avoiding any need of iterative solvers. The implicit discretization yields an elliptic equation on the pressure that can be solved for both ideal gas and general equation of state (EOS). A nested Newton method is used to solve the mildly nonlinear system for the pressure in case of nonlinear EOS. High order in time is granted by implicit-explicit (IMEX) time stepping, whereas a novel CWENO technique efficiently implemented in a dimension-by-dimension manner is developed for achieving high order in space for the discretization of explicit convective and viscous fluxes. A quadrature-free finite volume solver is then derived for the high order approximation of numerical fluxes. Central schemes with no dissipation of suitable order of accuracy are finally employed for the numerical approximation of the implicit terms. Consequently, the CFL-type stability condition on the maximum admissible time step is based only on the fluid velocity and not on the sound speed, so that the novel schemes work uniformly for all Mach numbers. Convergence and robustness of the proposed method are assessed through a wide set of benchmark problems involving low and high Mach number regimes, as well as inviscid and viscous flows.

Magnetism of Tsai-Type Quasicrystal Approximants

Magnetism of Tsai-type 1/1 approximants has been extensively investigated for the last decade. The experimental results on Tsai-type approximants are reviewed and the magnetic behaviors are classified and discussed in terms of the average electron-per-atom (e/a) ratio. The paramagnetic Curie-Weiss temperature Θp is found to oscillate in a universal manner as a function of the e/a ratio, and the magnetic order systematically changes with the variation in Θp. In addition, recent work shows that the 2/1 approximants also follow the trend observed for the 1/1 approximants, which implies that the magnetic phase diagram of the 1/1 approximants can be applicable to higher-order approximants including quasicrystals that are yet to be discovered.

Development of a four-node quadrilateral element-based high order numerical manifold method without linear dependency

A high-order numerical manifold method (HONMM) is developed using four-node quadrilateral (QUAD4) element to enhance accuracy and computational efficiency in solid mechanical problems. In the current study, the high-order global approximations are built by increasing the order of local approximations without facing the linear dependence (LD) problem, which is one of the main issues in the partition of unity (PU)-based methods with high-order approximations. To remove the LD problem, a new and simple scheme is proposed. Four problems are utilized to evaluate the efficiency of the QUAD4-based HONMM (QHONMM). A cantilever beam example is analyzed to compare different orders of QHONMM. Also, a block example with different loads and boundary conditions is used to investigate the sensitivity of the QHONMM results. Moreover, to demonstrate the capabilities of the QHONMM in dynamic analysis, free and forced vibrations of a simple beam under distributed and moving loads are analyzed. The results showed that using QUAD4 elements, the proposed QHONMM performs better than the conventional NMM in terms of accuracy and computational costs.

Asymptotic Expansion for Neural Network Operators of the Kantorovich Type and High Order of Approximation

In this paper, we study the rate of pointwise approximation for the neural network operators of the Kantorovich type. This result is obtained proving a certain asymptotic expansion for the above operators and then by establishing a Voronovskaja type formula. A central role in the above resuts is played by the truncated algebraic moments of the density functions generated by suitable sigmoidal functions. Furthermore, to improve the rate of convergence, we consider finite linear combinations of the above neural network type operators, and also in the latter case, we obtain a Voronovskaja type theorem. Finally, concrete examples of sigmoidal activation functions have been deeply discussed, together with the case of rectified linear unit (ReLu) activation function, very used in connection with deep neural networks.

Probabilistic analysis of dynamic stability for a rotating BDFG tapered beam with time-varying velocity and stochastic parameters

Taking the randomness of parameters into account, probabilistic dynamic stability analysis is carried out for a rotating tapered beam made of bidirectional functionally graded (BDFG) materials with time-dependent rotation speed. Hamilton’s principle is adopted to establish the equations of motion. The dynamic instability problem due to parametrical excitation is solved by Bolotin’s method with higher-order approximation. Considering parameter uncertainty, the stability-based system reliability model with multiple failure modes is clarified and established for probabilistic analysis. The active learning and Kriging-based system reliability (AK-SYS) method is used to estimate the stability-based system reliability of the rotating beam. In addition, an improved stopping criterion for an active learning procedure is introduced in the application of the AK-SYS method to accelerate iteration. Monte Carlo simulation is employed to verify the accuracy and efficiency of the proposed method. Finally, a numerical example is carried out for a parameter study, and the reliability-based sensitivities are investigated to rank the geometric and working parameters according to their importance to the system failure probability. Some useful information is provided for the design and use of rotating beam structures.

High-Order Mean-Field Approximations for Adaptive Susceptible-Infected-Susceptible Model in Finite-Size Networks

Exact solutions of epidemic models are critical for identifying the severity and mitigation possibility for epidemics. However, solving complex models can be difficult when interfering conditions from the real-world are incorporated into the models. In this paper, we focus on the generally unsolvable adaptive susceptible-infected-susceptible (ASIS) epidemic model, a typical example of a class of epidemic models that characterize the complex interplays between the virus spread and network structural evolution. We propose two methods based on mean-field approximation, i.e., the first-order mean-field approximation (FOMFA) and higher-order mean-field approximation (HOMFA), to derive the exact solutions to ASIS models. Both methods demonstrate the capability of accurately approximating the metastable-state statistics of the model, such as the infection fraction and network density, with low computational cost. These methods are potentially powerful tools in understanding, mitigating, and controlling disease outbreaks and infodemics.

Interpolation of Hermite data by clamped Minkowski Pythagorean hodograph B-spline curves

In this paper, the problem of Hermite interpolation by clamped Minkowski Pythagorean hodograph (MPH) B-spline curves is considered. Using the properties of B-splines, our intention is to use the MPH curves of degrees lower than in algorithms designed before. Special attention is devoted to C1∕C2 Hermite interpolation by MPH B-spline cubics/quintics. The resulting interpolants are obtained by exploiting properties of B-spline basis functions and via solving special quadratic and linear equations in Clifford algebra Cℓ2,1. All the presented algorithms are purely symbolic. The results are confirmed by several applications, in particular we use them to generate an approximate conversion of a given analytic curve to MPH B-spline curve with a high order of approximation, then to an efficient approximation of the medial axis transform of a planar domain leading to NURBS representation of the (trimmed) offsets of the domain boundaries, and to skinning of systems of circles in plane.

Bulk electronic structure of high-order quaternary approximants

The authors use hard x-ray photoelectron spectroscopy to show the existence of a pseudogap at the Fermi level in high-order quaternary approximants, Al-Pd-Mo-Fe and Al-Pd-Cr-Fe.

Parallel FFT algorithms for high-order approximations on three-dimensional compact stencils

The recent development of multicore technologies on modern desktop computers makes parallelization of the proposed numerical approaches a priority in algorithmic research. The main performance improvement of personal computers in the upcoming years will be made based on the increasing number of cores on modern CPUs. This shifts the focus of algorithmic research from the development of sequential numerical methods to parallel methodology. This paper presents an efficient parallel direct algorithm with near-optimal complexity for the compact fourth and sixth-order approximation of the three-dimensional Helmholtz equations (Turkel et al., 2013) with the problem coefficient depending on only one of the coordinate directions. The developed method is based on a combination of the separation of variables technique and a Fast Fourier Transform (FFT) type method. Similar direct solvers for the lower-order approximations of the two and three-dimensional Helmholtz equation were considered in several previous publications by the authors and other researchers (see, e.g. Gryazin et al. (2000); Gryazin (2014); Elman and O’Leary (1998); Elman and O’Leary (1999); Toivanen and Wolfmayr (2020)). The authors also consider a generalization of the presented algorithm to the solution of a wide class of linear systems obtained from approximation on the compact 27-point three-dimensional stencils on the rectangular grids with similar requirements on the stencil coefficients. The general restrictions on the coefficients in the considered class of compact schemes are developed and presented. This class includes the second, fourth and sixth-order compact approximation schemes for the three-dimensional Helmholtz equation considered in this paper and our previous publications (Gryazin et al., 2000; Gryazin, 2014; Gryazin, 2014). As an example of the diversity of applications of the developed general method, the direct parallel implementation of a compact fourth-order approximation scheme for a convection–diffusion equation is considered. Another goal of this paper is to investigate the scalability of the proposed technique in the case of a large linear system using different parallel programming extensions. The results of the implementation of this method in OpenMP, MPI and hybrid programming environments on the multicore computers and multiple node clusters are presented and discussed. The results demonstrate the high efficiency of the proposed direct solvers for many important applications on the structured grid with the corresponding 27-diagonal matrices of sizes up to 1011 by 1011.

Method of moments with a choice of special basic functions for fourth-order partial differential equations

This paper presents an approximation method to solve the boundary value problem (BVP) for partial differential equations (PDEs) of a kind of the fourth order with the idea of discretization of a spatial variable by using the method of moments with a choice of special basic functions. A system of ordinary differential equations (ODEs) is obtained by multiplying the original equation by some auxiliary functions, followed by interpolation and integration over the spatial variable. Newton-Stirling, Hermite-Birkhoff and Hermite interpolations are flexibly applied to internal and pre-boundary nodes. In addition, boundary conditions are automatically satisfied without being approximated separately as in classical numerical methods (for example, method of grids, method of lines). Thus, the proposed schemes have a higher order of approximation. The main goal of the work is the construction of basic functions for a fourth-order differential operator and a possible increase in the order of the remainder term when passing to difference equations.