Closed-form expressions for the finite difference approximations of first and higher derivatives based on Taylor series

Numerical differentiation formulas based on interpolating polynomials, operators and lozenge diagrams can be simplified to one of the finite difference approximations based on Taylor series. In this paper, we have presented closed-form expressions of these approximations of arbitrary order for first and higher derivatives. A comparison of the three types of approximations is given with an ideal digital differentiator by comparing their frequency responses. The comparison reveals that the central difference approximations can be used as digital differentiators, because they do not introduce any phase distortion and their amplitude response is closer to that of an ideal differentiator. It is also observed that central difference approximations are in fact the same as maximally flat digital differentiators. In the appendix, a computer program, written in MATHEMATICA is presented, which can give the approximation of any order to the derivative of a function at a certain mesh point.     

A polynomial approximation for arbitrary functions

We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, for approximating arbitrary functions. We show that the polynomial coefficients in the Legendre expansion, and thus, the whole series, converge to zero much more rapidly compared to those in the Taylor expansion of the same order. Furthermore, using numerical analysis with a sixth-order polynomial expansion, we demonstrate that the Legendre polynomial approximation yields an error at least an order of magnitude smaller than that of the analogous Taylor series approximation. This strongly suggests that Legendre expansions, instead of Taylor expansions, should be used when global accuracy is important.     

Recurrence Relations for the Coefficients in Ultraspherical Series Solutions of Ordinary Differential Equations

A method is presented for obtaining recurrence relations for the coefficients in ultraspherical series of linear differential equations. This method applies Doha's method (1985) to generate polynomial approximations in terms of ultraspherical polynomials of $y(zx), -1\leq x\leq 1,z\in C,|z|\leq 1$, where y is a solution of a linear differential equation. In particular, rational approximations of $y(z)$ result if $x$ is set equal to unity. Two numerical examples are given to illustrate the application of the method to first and second order differential equations. In general, the rational approximations obtained by this method are better than the corresponding polynomial approximations, and compare favourably with Pade approximants.     

On Finite Element Approximations to a Shape Gradient Flow in Shape Optimization of Elliptic Problems

Shape gradient flows are widely used in numerical shape optimization algorithms. We investigate the accuracy and effectiveness of approximate shape gradients flows for shape optimization of elliptic problems. We present convergence analysis with a priori error estimates for finite element approximations of shape gradient flows associated with a distributed or boundary expression of Eulerian derivative. Numerical examples are presented to verify theory and show that using the volume expression is effective for shape optimization with Dirichlet and Neumann boundary conditions.     

非线性发展方程的Taylor展开方法

提出了积分非线性发展方程的新方法,即Taylor展开方法．标准的Galerkin方法可以看作0-阶Taylor展开方法,而非线性Galerkin方法可以看作1-阶修正Taylor展开方法A·D2此外,证明了数值解的存在性及其收敛性．结果表明,在关于严格解的一些正则性假设下,较高阶的Taylor展开方法具有较高阶的收敛速度．最后,给出了用Taylor展开方法求解二维具有非滑移边界条件Navier-Stokes方程的具体例子．     

Optimization via Chebyshev polynomials

This paper presents for the first time a robust exact line-search method based on a full pseudospectral (PS) numerical scheme employing orthogonal polynomials. The proposed method takes on an adaptive search procedure and combines the superior accuracy of Chebyshev PS approximations with the high-order approximations obtained through Chebyshev PS differentiation matrices. In addition, the method exhibits quadratic convergence rate by enforcing an adaptive Newton search iterative scheme. A rigorous error analysis of the proposed method is presented along with a detailed set of pseudocodes for the established computational algorithms. Several numerical experiments are conducted on one- and multi-dimensional optimization test problems to illustrate the advantages of the proposed strategy.     

Quadratic convergence for cell-centered grids

This work discusses some of the convergence properties of approximations defined on standard cell-centered finite difference grids. It is shown that the order of convergence is quadratic in the grid spacing for both uniform and nonuniform grids. This order of convergence cannot be improved upon, even if uniform point-distributed grids are used. It is concluded that order of convergence arguments do not favor point-distributed grid construction over the more physically reasonable cell-centered construction. The techniques used are elementary and rely entirely on Taylor series expansions. Other applications of these techniques, such as to local grid refinement, are indicated.     

Numerical Simulation of Incompressible Flows Within Simple Boundaries. I. Galerkin (Spectral) Representations

Galerkin (spectral) methods are explored for the numerical simulation of incompressible flows within simple boundaries. A major part of the paper is devoted to the development of transform methods for efficient simulation of flows in box geometries with periodic and free-slip boundary conditions. Techniques for incorporating known symmetries and invariances into transform methods are illustrated for the Taylor-Green vortex. Galerkin methods for accurate and efficient representation of rigid no-slip boundary conditions are also explained. A class of pseudospectral approximations is introduced in order to handle more complicated dynamical interactions in more complicated geometries. Later papers in this series will demonstrate the important advantages of spectral methods over finite-difference methods for simulation of many of the flows of current interest and will present specific numerical results for various transition and turbulent flows.     

Least‐square collocation and Lagrange multipliers forTaylor meshless method

A recently proposed meshless method is discussed in this article. It relies on Taylor series, the shape functions being high degree polynomials deduced from the Partial Differential Equation (PDE). In this framework, an efficient technique to couple several polynomial approximations has been presented in (Tampango, Potier‐Ferry, Koutsawa, Tiem, Int. J. Numer. Meth. Eng. vol. 95 (2013) pp. 1094–1112): the boundary conditions were applied using the least‐square collocation and the interface was coupled by a bridging technique based on Lagrange multipliers. In this article, least‐square collocation and Lagrange multipliers are applied for boundary conditions, respectively, and least‐square collocation is revisited to account for the interface conditions in piecewise resolutions. Various combinations of these two techniques have been investigated and the numerical results prove their effectiveness to obtain very accurate solutions, even for large scale problems.     

A Chebyshev interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new interval analysis method for the dynamic response of nonlinear systems with uncertain-but-bounded parameters using Chebyshev polynomial series. Interval model can be used to describe nonlinear dynamic systems under uncertainty with low-order Taylor series expansions. However, the Taylor series-based interval method can only suit problems with small uncertain levels. To account for larger uncertain levels, this study introduces Chebyshev series expansions into interval model to develop a new uncertain method for dynamic nonlinear systems. In contrast to the Taylor series, the Chebyshev series can offer a higher numerical accuracy in the approximation of solutions. The Chebyshev inclusion function is developed to control the overestimation in interval computations, based on the truncated Chevbyshev series expansion. The Mehler integral is used to calculate the coefficients of Chebyshev polynomials. With the proposed Chebyshev approximation, the set of ordinary differential equations (ODEs) with interval parameters can be transformed to a new set of ODEs with deterministic parameters, to which many numerical solvers for ODEs can be directly applied. Two numerical examples are applied to demonstrate the effectiveness of the proposed method, in particular its ability to effectively control the overestimation as a non-intrusive method.     

From formal numerical solutions of elliptic PDE's to the true ones

We propose a discretization scheme for a numerical solution of elliptic PDE's, based on local representation of functions, by their Taylor polynomials (jets). This scheme utilizes jet calculus to provide a very high order of accuracy for a relatively small number of unknowns involved.

     

Boundary slip phenomena in a binary gas mixture

The slip phenomena in gas mixtures are of fundamental significance in the specification of boundary conditions for flows in the slip regime. In a recent paper, new explicit results for the slip coefficients appropriate to binary gas mixtures were reported. The present work being reported extends the previous work to a higher level of accuracy by involving a higher order Chapman-Enskog expansion. In particular, new expressions for the slip coefficients are presented which are applicable for arbitrary models of the intermolecular interaction. Limiting expressions for the slip coefficients are given (for a simple gas) and the accuracy of the theory is discussed. Numerical calculations of the slip coefficients for different binary gas mixtures using the first and second order Chapman-Enskog approximations and the rigid sphere and Lennard-Jones (12-6) potential models have been carried out. The thermal creep and diffusion slip coefficients are found to be sensitive to the order of the approximation and to the potential model used. A comparison of the new higher order results with some of our previously obtained experimental data for the thermal transpiration effect has also been carried out and shows excellent agreement between the theory and the experiments which confirms the accuracy of the theory.     

外推法对奇异摄动问题数值解的应用

在本文中讨论了外推法对椭圆一抛物奇异摄动问题数值解的应用,提高了解的精度,估出了精度的阶数,并对文[1]中的一致收敛性在附录中给出证明.     

Approximating exponential and logarithmic functions using polynomial interpolation

This article takes a closer look at the problem of approximating the exponential and logarithmic functions using polynomials. Either as an alternative to or a precursor to Taylor polynomial approximations at the precalculus level, interpolating polynomials are considered. A measure of error is given and the behaviour of the error function is analysed. The results of interpolating polynomials are compared with those of Taylor polynomials.     

A matrix method for fractional Sturm-Liouville problems on bounded domain

A matrix method for the solution of direct fractional Sturm-Liouville problems (SLPs) on bounded domains is proposed where the fractional derivative is defined in the Riesz sense. The scheme is based on the application of the Galerkin spectral method of orthogonal polynomials. The order of convergence of the eigenvalue approximations with respect to the matrix size is studied. Some numerical examples that confirm the theory and prove the competitiveness of the approach are finally presented.     

A Poisson–Charlier approximation for nonstationary queues

In this paper, we develop a new approximation for nonstationary multiserver queues with abandonment. Our method uses the Poisson–Charlier polynomials, which are a discrete orthogonal polynomial sequence that is orthogonal with respect to the Poisson distribution. We show that by appealing to the Poisson–Charlier polynomials that we can estimate the mean, variance, and probability of delay of our nonstationary queueing system with good accuracy. Lastly, we provide a numerical example that illustrates that our approximations are effective.     

符号函数的Legendre非线性逼近

1引言许多数学和物理工作者研究了逼近形式正交多项式级数的具有较好收敛性的非线性方法,如文献[2-5,9]．这些非线性逼近方法的一个共同点是使用了线性级数中正交多项式的母函数．众所周知,的符号函数具有很多的应用,如文献[7]利用符号函数的积分表示来分析相联存储器的回想过程．文献[1]及其中所引用的一些文献为了获得交迭格Dirac算子,讨论了符号函数的有理逼近和连分式展开．在本文中,我们研究符号函数的Lengendre     

OPTIMAL ERROR ESTIMATES OF THE PARTITION OF UNITY METHOD WITH LOCAL POLYNOMIAL APPROXIMATION SPACES

In this paper, we provide a theoretical analysis of the partition of unity finite elementmethod (PUFEM), which belongs to the family of meshfree methods. The usual erroranalysis only shows the order of error estimate to the same as the local approximations[12].Using standard linear finite element base functions as partition of unity and polynomials aslocal approximation space, in 1-d case, we derive optimal order error estimates for PUFEMinterpolants. Our analysis show that the error estimate is of one order higher than thelocal approximations. The interpolation error estimates yield optimal error estimates forPUFEM solutions of elliptic boundary value problems.     

Elzaki Transform Based Accelerated Homotopy Perturbation Method for Multi-dimensional Smoluchowski"s Coagulation and Coupled Coagulation-fragmentation Equations

This article aims to establish a semi-analytical approach based on the homotopy perturbation method (HPM) to find the closed form or approximated solutions for the population balance equations such as Smoluchowski"s coagulation, fragmentation, coupled coagulation-fragmentation and bivariate coagulation equations. An accelerated form of the HPM is combined with the Elzaki transformation to improve the accuracy and efficiency of the method. One of the significant advantages of the technique lies over the classic numerical methods as it allows solving the linear and non-linear differential equations without discretization. Further, it has benefits over the existing semi-analytical techniques such as Adomian decomposition method (ADM), optimized decomposition method (ODM), and homotopy analysis method (HAM) in the sense that computation of Adomian polynomials and convergence parameters are not required. The novelty of the scheme is shown by comparing the numerical findings with the existing results obtained via ADM, HPM, HAM and ODM for non-linear coagulation equation. This motivates us to extend the scheme for solving the other models mentioned above. The supremacy of the proposed scheme is demonstrated by taking several numerical examples for each problem. The error between exact and series solutions provided in graphs and tables show the accuracy and applicability of the method. In addition to this, convergence of the series solution is also the key attraction of the work.     

A high accuracy hybrid method for two-dimensional Navier–Stokes equations

A dual-mesh hybrid numerical method is proposed for high Reynolds and high Rayleigh number flows. The scheme is of high accuracy because of the use of a fourth-order finite-difference scheme for the time-dependent convection and diffusion equations on a non-uniform mesh and a fast Poisson solver DFPS2H based on the HODIE finite-difference scheme and algorithm HFFT [R.A. Boisvert, Fourth order accurate fast direct method for the Helmholtz equation, in: G. Birkhoff, A. Schoenstadt (Eds.), Elliptic Problem Solvers II, Academic Press, Orlando, FL, 1984, pp. 35–44] for the stream function equation on a uniform mesh. To combine the fast Poisson solver DFPS2H and the high-order upwind-biased finite-difference method on the two different meshes, Chebyshev polynomials have been used to transfer the data between the uniform and non-uniform meshes. Because of the adoption of a hybrid grid system, the proposed numerical model can handle the steep spatial gradients of the dependent variables by using very fine resolutions in the boundary layers at reasonable computational cost. The successful simulation of lid-driven cavity flows and differentially heated cavity flows demonstrates that the proposed numerical model is very stable and accurate within the range of applicability of the governing equations.