Barycentric Rational Interpolation Research Articles

Algorithm 1048: A C++ Class for Robust Linear Barycentric Rational Interpolation

Barycentric rational interpolation is a recent interpolation method with several favourable properties. In this article, we present the BRI class, which features a new C++ class template that contains all variables and functions related to linear barycentric rational interpolation. While several methods exist to evaluate a barycentric rational interpolant, the class is designed to autonomously select the best method to use on a case-by-case basis, as it takes into account the latest results regarding the efficiency and numerical stability of barycentric rational interpolation [ 15 ]. Moreover, we describe a new technique that makes the code robust and less prone to overflow and underflow errors. In addition to the standard C++ data types, the BRI template variables can also be defined with arbitrary precision because the BRI class is compatible with the Multiple Precision Floating-Point Reliable (MPFR) library [ 14 ].

Meshless Barycentric Rational Interpolation Method for Solving Nonlinear Stochastic Fractional Integro-Differential Equations

Meshless Barycentric Rational Interpolation Method for Solving Nonlinear Stochastic Fractional Integro-Differential Equations

Stability Analysis of Milling Based on the Barycentric Rational Interpolation Differential Quadrature Method

Chatter causes great damage to the machining process, and the selection of appropriate process parameters through chatter stability analysis is of great significance for achieving chatter-free machining. This article proposes a milling stability analysis method based on the barycentric rational interpolation differential quadrature method (DQM). The dynamics of the milling process considering the regeneration effect is first modelled as a time-delay differential equation (DDE). When adjacent pitch angles of the milling cutter are symmetric, the milling dynamic equation contains a single time delay. Otherwise, when adjacent pitch angles are asymmetric, the dynamic equation contains multiple time delays. The barycentric rational interpolation DQM is then used to approximate the differential and delay terms of the milling dynamics equation, and to construct a state transition matrix between adjacent milling periods. Finally, the chatter stability lobe diagram (SLD) is obtained based on the Floquet theory. According to the SLD, the appropriate spindle speed can be selected to obtain the maximum stable axial depth of cutting, thereby effectively improving the material removal rate. The accuracy and efficiency of the proposed method have been validated by two widely used milling models, and the results show that the proposed method has good accuracy and computational efficiency.

A class of reducible quadrature rules for the second-kind Volterra integral equations using barycentric rational interpolation

Reducible quadrature rules constitute a well-established class of direct quadrature methods for approximating solutions to Volterra integral equations. Unlike interpolatory quadrature formulae, the reducible quadrature rule can be constructed without the need for additional calculations of moments. This paper investigates the reducible quadrature rule by employing barycentric rational interpolation to solve the underlying initial value problem and analyzes its convergence and stability. It is found that these quadrature rules exhibit high-order convergence rates accompanied by extensive stability regions. Several numerical illustrations are provided to verify the theoretical results.

Higher order class of finite difference method for time-fractional Liouville-Caputo and space-Riesz fractional diffusion equation

In this paper, a class of finite difference method (FDM) is designed for solving the timefractional Liouville-Caputo and space-Riesz fractional diffusion equation. For this purpose, the fractional linear barycentric rational interpolation method (FLBRI) is adopted to discretize the Liouville-Caputo derivative in the time direction as well as the second order revised fractional backward difference formulae 2 (RFBDF2) is employed in the space direction. The energy method is used to prove unconditionally stability and convergence analysis of the proposed method. Eventually, it is concluded that the proposed method is convergent with the order O(h? t + h2x ), where ht and hx are the temporal and the spatial step sizes respectively, and 1 ? ? ? 7 is the order of accuracy in the time direction. Finally, the presented numerical experiment confirms the theoretical analysis, the high accuracy and efficiency of the offered method.

The barycentric rational predictor-corrector schemes for Volterra integral equations

This paper introduces a family of barycentric rational predictor-corrector schemes based on the Floater–Hormann family of linear barycentric rational interpolants (LBRIs) for the numerical solution of classical systems of second-kind Volterra integral equations. Also, we introduce a family of LBRI-based predictor-corrector starting procedures that is essentially explicit and whose order of convergence can be as high as that of the main method. Numerical tests verify the theoretical results on the convergence order and stability and illustrate the efficiency and power of the developed family of methods in solving stiff equations.

Combination of discrete technique on graded meshes with barycentric rational interpolation for solving a class of time-dependent partial integro-differential equations with weakly singular kernels

Combination of discrete technique on graded meshes with barycentric rational interpolation for solving a class of time-dependent partial integro-differential equations with weakly singular kernels

Collocation methods based on barycentric rational interpolation for Volterra integro‐differential equations with weakly singular kernels

This paper is concerned with the numerical solution of Volterra integro‐differential equations with weakly singular kernels. A smoothing transformation is first used to convert the original equation to a new equation with better regularity. A collocation method based on barycentric rational interpolation is introduced. The convergence and supercovergence of the numerical solution are studied in detail. Some numerical results are illustrated to confirm the theoretical prediction of the rate of convergence and superconvergence.

A combination of the quasilinearization method and linear barycentric rational interpolation to solve nonlinear multi-dimensional Volterra integral equations

In this paper, an iterative scheme including a combination of the quasilinearization technique and multi-dimensional linear barycentric rational interpolation is applied to solve nonlinear multi-dimensional Volterra integral equations. First, employing the quasilinearization method, the nonlinear multi-dimensional Volterra integral equation is reduced to a sequence of linear Volterra integral equations. Under appropriate assumptions, the constructed iterative sequence is uniformly convergent to the unique solution of the problem. In general, finding an analytical solution for linear Volterra integral equations is impossible. Hence, in each iteration, using a collocation method based on multi-dimensional barycentric rational basis functions, the solution to the linear integral equation is approximated. The quadratic convergence of the quasilinearization approach and the error estimation of the combined method are investigated theoretically. In the end, the efficiency and the validity of this method are illustrated with some numerical examples and compared with those of the existing numerical methods.

Barycentric rational interpolation method for solving time-dependent fractional convection-diffusion equation

<abstract><p>The time-dependent fractional convection-diffusion (TFCD) equation is solved by the barycentric rational interpolation method (BRIM). Since the fractional derivative is the nonlocal operator, we develop a spectral method to solve the TFCD equation to get the coefficient matrix as a full matrix. First, the fractional derivative of the TFCD equation is changed to a nonsingular integral from the singular kernel to a density function. Second, efficient quadrature of the new Gauss formula are constructed to simply compute it. Third, matrix equation of discrete the TFCD equation is obtained by the unknown function replaced by a barycentric rational interpolation basis function. Then, the convergence rate of BRIM is proved. Finally, a numerical example is given to illustrate our result.</p></abstract>

Barycentric rational interpolation method for solving KPP equation

<abstract><p>In this paper, we seek to solve the Kolmogorov-Petrovskii-Piskunov (KPP) equation by the linear barycentric rational interpolation method (LBRIM). As there are non-linear parts in the KPP equation, three kinds of linearization schemes, direct linearization, partial linearization, Newton linearization, are presented to change the KPP equation into linear equations. With the help of barycentric rational interpolation basis function, matrix equations of three kinds of linearization schemes are obtained from the discrete KPP equation. Convergence rate of LBRIM for solving the KPP equation is also proved. At last, two examples are given to prove the theoretical analysis.</p></abstract>

Barycentric rational interpolation method for solving fractional cable equation

<abstract><p>A fractional cable (FC) equation is solved by the barycentric rational interpolation method (BRIM). As the fractional derivative is a nonlocal operator, we develop a spectral method to solve the FC equation to get the coefficient matrix as the full matrix. First, the fractional derivative of the FC equation is changed to a nonsingular integral from the singular kernel to the density function. Second, an efficient quadrature of a new Gauss formula is constructed to compute it simply. Third, a matrix equation of the discrete FC equation is obtained by the unknown function replaced by a barycentric rational interpolation basis function. Then, convergence rate for FC equation of the BRIM is derived. At last, a numerical example is given to illustrate our results.</p></abstract>

Linear barycentric rational interpolation method for solving Kuramoto-Sivashinsky equation

<abstract><p>The Kuramoto-Sivashinsky (KS) equation being solved by the linear barycentric rational interpolation method (LBRIM) is presented. Three kinds of linearization schemes, direct linearization, partial linearization and Newton linearization, are presented to get the linear equation of the Kuramoto-Sivashinsky equation. Matrix equations of the discrete Kuramoto-Sivashinsky equation are also given. The convergence rate of LBRIM for solving the KS equation is also proved. At last, two examples are given to prove the theoretical analysis.</p></abstract>

A kernel-independent uniform fast multipole method based on barycentric rational interpolation

A kernel-independent uniform fast multipole method based on barycentric rational interpolation

A numerical study of the Burgers’ and Fisher’s equations using barycentric interpolation method

PurposeThe purpose of the method is to develop a numerical method for the solution of nonlinear partial differential equations.Design/methodology/approachA new numerical approach based on Barycentric Rational interpolation has been used to solve partial differential equations.FindingsA numerical technique based on barycentric rational interpolation has been developed to investigate numerical simulation of the Burgers’ and Fisher’s equations. Barycentric interpolation is basically a variant of well-known Lagrange polynomial interpolation which is very fast and stable. Using semi-discretization for unknown variable and its derivatives in spatial direction by barycentric rational interpolation, we get a system of ordinary differential equations. This system of ordinary differential equation’s has been solved by applying SSP-RK43 method. To check the efficiency of the method, computed numerical results have been compared with those obtained by existing methods. Barycentric method is able to capture solution behavior at small values of kinematic viscosity for Burgers’ equation.Originality/valueTo the best of the authors’ knowledge, the method is developed for the first time and validity is checked by stability and error analysis.

High-order finite difference method based on linear barycentric rational interpolation for Caputo type sub-diffusion equation

The main aim of this paper is to develop a class of high-order finite difference method for the numerical solution of Caputo type time-fractional sub-diffusion equation. In the time direction, the Caputo derivative is discretized by employing a numerical technique based on the fractional linear barycentric rational interpolation method (FLBRI). The stability properties and the convergence analysis of the proposed scheme are considered based on the energy method. As a result, the order of convergence is O(Δt)p+(Δx)2 for 1≤p≤7, in which Δt is the temporal step size, Δx is the spatial step size, and p is the order of accuracy in the time direction. Finally, numerical experiment is provided to show that the results confirm the theoretical analysis, perfectly.

Extended SDBDF-Type Methods Based on Linear Barycentric Rational Interpolants for ODEs

Extended SDBDF-Type Methods Based on Linear Barycentric Rational Interpolants for ODEs

Numerical simulation of two‐dimensional and three‐dimensional generalizedKlein–Gordon–Zakharovequations with power law nonlinearity via a meshless collocation method based on barycentric rational interpolation

AbstractThis study presents numerical simulations of generalized two‐dimensional (2D) and three‐dimensional (3D) Klein–Gordon–Zakharov (KGZ) equations with power law nonlinearity, which are coupled nonlinear partial differential equations. A meshless collocation method based on barycentric rational interpolation is developed for space variable of the KGZ equations. For time discretization, an explicit low storage fourth order Runge Kutta method is proposed after transforming KGZ equations to system of ordinary differential equations by introducing auxiliary variables.L∞andL2error norms for some test problems are computed. Obtained numerical results and comparisons with finite element methods indicate that barycentric rational interpolation method is an efficient method for solving multidimensional generalized KGZ system numerically.

Explicit methods based on barycentric rational interpolants for solving non-stiff Volterra integral equations

For their high accuracy and good stability properties, implicit numerical methods are widely used for solving Volterra integral equations, while, in order to save computational effort, explicit algorithms are preferred in the case of non-stiff problems. In this paper, highly accurate explicit methods based on the Floater–Hormann family of linear barycentric rational interpolants are presented. The order of convergence is obtained in terms of the parameters of the methods. Moreover, the linear stability properties with respect to both the basic and convolution test equations are analyzed in detail. Numerical experiments are discussed in order to validate the theoretical results and illustrate the efficiency and power of the methods applied to non-stiff and mildly stiff problems.

Numerical Simulation of the Temperature in a Train Brake Disc by Barycentric Rational Interpolation Collocation Method

Numerical Simulation of the Temperature in a Train Brake Disc by Barycentric Rational Interpolation Collocation Method