Lobatto Points Research Articles

Stability analysis and highly accurate numerical approximation of Fisher’s equations using pseudospectral method

In this article, we present a new pseudospectral method to approximate the solution of one- and two-dimensional Fisher’s equations, which is a prototype of nonlinear reaction–diffusion equations. The proposed method is based on Chebyshev–Gauss–Lobatto points and employed collocation in both spatial and time direction to study of several travelling wave solutions which evolves into a shock like wavefront and change the shape to some wave pattern. The stability analysis of the proposed method for Fisher’s equation is presented. The numerical results show highly accurate and stable for different values of the reaction rate coefficients. Some model examples of one- and two-dimensional Fisher’s equations are tested and detailed comparison of the proposed method with various other methods are also given.

An $h$-$p$ version of the Chebyshev spectral collocation method for Volterra integro-differential equations with vanishing delays

We propose a spectral collocation method based on the Chebyshev–Gauss–Lobatto points for linear Volterra integro-differential equations with vanishing delays. We derive an h-p version a-priori error estimate in the H1-norm that is completely explicit in the local time steps and the local polynomial degrees. Numerical experiments are provided to confirm the theoretical results.

Legendre-tau-Galerkin and spectral collocation method for nonlinear evolution equations

A Legendre-tau-Galerkin method is developed for nonlinear evolution problems and its multiple interval form is also considered. The Legendre tau method is applied in time and the Legendre/Chebyshev–Gauss–Lobatto points are adopted to deal with the nonlinear term. By taking appropriate basis functions, it leads to a simple discrete equation. The proposed method enables us to derive optimal error estimates in L2-norm for the Legendre collocation under the two kinds of Lipschitz conditions, respectively. Our method is also applied to the numerical solutions of some nonlinear partial differential equations by using the Legendre Galerkin and Chebyshev collocation in spatial discretization. Numerical examples are given to show the efficiency of the methods.

A reduced computational matrix approach with convergence estimation forsolving model differential equations involving specific nonlinearities ofquartic type

This study aims to efficiently solve model differential equations involving specific nonlinearities of quartic type by proposing a reduced computational matrix approach based on the generalized Mott polynomial. This method presents a reduced matrix expansion of the generalized Mott polynomial with the parameter-$\alpha$, matrix equations, and Chebyshev--Lobatto collocation points. The simplicity of the method provides fast computation while eliminating an algebraic system of nonlinear equations, which arises from the matrix equation. The method also scrutinizes the consistency of the solutions due to the parameter-$\alpha$. The oscillatory behavior of the obtained solutions on long time intervals is simulated via a coupled methodology involving the proposed method and Laplace-Padé technique. The convergence estimation is established via residual function. Numerical and graphical results are indicated to discuss the validity and efficiency of the method.

Superconvergence of $$C^0$$-$$Q^k$$ Finite Element Method for Elliptic Equations with Approximated Coefficients

We prove that the superconvergence of $$C^0$$-$$Q^k$$ finite element method at the Gauss–Lobatto quadrature points still holds if variable coefficients in an elliptic problem are replaced by their piecewise $$Q^k$$ Lagrange interpolants at the Gauss–Lobatto points in each rectangular cell. In particular, a fourth order finite difference type scheme can be constructed using $$C^0$$-$$Q^2$$ finite element method with $$Q^2$$ approximated coefficients.

A novel Chebyshev-wavelet-based approach for accurate and fast prediction of milling stability

Currently, semi-analytical stability analysis methods for milling processes focus on improving prediction accuracy and simultaneously reducing computing time. This paper presents a Chebyshev-wavelet-based method for improved milling stability prediction. When including regenerative effect, the milling dynamics model can be concluded as periodic delay differential equations, and is re-presented as state equation forms via matrix transformation. After divide the period of the coefficient matrix into two subintervals, the forced vibration time interval is mapped equivalently to the definition interval of the second kind Chebyshev wavelets. Thereafter, the explicit Chebyshev–Gauss–Lobatto points are utilized for discretization. To construct the Floquet transition matrix, the state term is approximated by finite series second kind Chebyshev wavelets, while its derivative is acquired with a simple and explicit operational matrix of derivative. Finally, the milling stability can be semi-analytically predicted using Floquet theory. The effectiveness and superiority of the presented approach are verified by two benchmark milling models and comparisons with the representative existing methods. The results demonstrate that the presented approach is highly accurate, fast and easy to implement. Meanwhile, it is shown that the presented approach achieves high stability prediction accuracy and efficiency for both large and low radial-immersion milling operations.

Space–Time Spectral Collocation Method for Solving Burgers Equations with the Convergence Analysis

This article deals with a numerical approach based on the symmetric space-time Chebyshev spectral collocation method for solving different types of Burgers equations with Dirichlet boundary conditions. In this method, the variables of the equation are first approximated by interpolating polynomials and then discretized at the Chebyshev–Gauss–Lobatto points. Thus, we get a system of algebraic equations whose solution is the set of unknown coefficients of the approximate solution of the main problem. We investigate the convergence of the suggested numerical scheme and compare the proposed method with several recent approaches through examining some test problems.

An efficient spectral-Galerkin method for solving two-dimensional nonlinear system of advection–diffusion–reaction equations

The spectral Legendre–Galerkin method for solving a two-dimensional nonlinear system of advection–diffusion–reaction equations on a rectangular domain is presented and compared with analytical solution. The proposed method is based on the Legendre–Galerkin formulation for the linear terms and computation of the nonlinear terms in the Chebyshev–Gauss–Lobatto points. The main difference of the spectral Legendre–Galerkin method presented in the current paper with the classic Legendre–Galerkin method is in treating the nonlinear terms and imposing boundary conditions. Indeed, in the spectral Legendre–Galerkin method the nonlinear terms are efficiently handled using the Chebyshev–Gauss–Lobatto points and also the boundary conditions are imposed strongly as collocation methods. Combination of the proposed method with a semi-implicit time integration method such as the Leapfrog–Crank–Nicolson scheme leads to reducing the complexity of computations and obtaining a linear algebraic system of equations. Efficiency and spectral accuracy of the proposed method are demonstrated numerically by some examples.

Mixed Spectral-Element Method for the Waveguide Problem With Bloch Periodic Boundary Conditions

The mixed spectral-element method (MSEM) is applied to solve the waveguide problem with the Bloch periodic boundary condition (BPBC). Based on the BPBC for the original Helmholtz equation and the periodic boundary condition (PBC) for the equivalent but modified Helmholtz equation, two equivalent mixed variational formulations are applied for the MSEM. Unlike the traditional finite-element method and spectral-element method (SEM), both these mixed SEM schemes are completely free of spurious modes because of their use of the Gauss’ law and the curl-conforming vector basis functions structured by the Gauss–Legendre–Lobatto points. A simple implementation method is used to deal with the BPBC and the PBC for the mixed variational formulations so that both schemes can save computational costs over the traditional methods. Several numerical results are also provided to verify that both schemes are free of spurious modes and have high accuracy with the propagation constants.

Superconvergence of the direct discontinuous Galerkin method for a time‐fractional initial‐boundary value problem

AbstractA time‐fractional reaction–diffusion initial‐boundary value problem with periodic boundary condition is considered on Q ≔ Ω × [0, T], where Ω is the interval [0, l]. Typical solutions of such problem have a weak singularity at the initial time t = 0. The numerical method of the paper uses a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh, with piecewise polynomials of degree k ≥ 2. In the temporal direction we use the L1 approximation of the Caputo derivative on a suitably graded mesh. We prove that at each time level of the mesh, our L1‐DDG solution is superconvergent of order k + 2 in L2(Ω) to a particular projection of the exact solution. Moreover, the L1‐DDG solution achieves superconvergence of order (k + 2) in a discrete L2(Q) norm computed at the Lobatto points, and order (k + 1) superconvergence in a discrete H1(Q) seminorm at the Gauss points; numerical results show that these estimates are sharp.

Convergence and error analysis of a spectral collocation method for solving system of nonlinear Fredholm integral equations of second kind

This paper presents a new numerical approximation method to solve a system of nonlinear Fredholm integral equations of second kind. Spectral collocation method and their properties are applied to determine the general solution procedure for nonlinear Fredholm integral equations (FIEs). The convergence and error analysis of spectral collocation method are incorporated for the given nonlinear model. Legendre–Gauss–Lobatto (LGL) points are used as collocation points with various Legendre–Gauss quadrature with weight functions. The use of Legendre polynomials, together with the Gauss quadrature collocation points is well known for the accurate approximations that converge exponentially. Finally, we validate our theoretical results with a number of numerical examples, which further enhance the efficiency of our proposed scheme.

Acoustic waves scattered by elastic waveguides using a spectral approach with a 2.5D coupled boundary-finite element method

This work presents a two-and-a-half dimensional (2.5D) spectral formulation based on the finite element method (FEM) and the boundary element method (BEM) to study wave propagation in acoustic and elastic waveguides. The analysis involved superposing two dimensional (2D) problems with different longitudinal wavenumbers. A spectral finite element (SFEM) is proposed to represent waveguides in solids with arbitrary cross-section. Moreover, the BEM is extended to its spectral formulation (SBEM) to study unbounded fluid media and acoustic enclosures. Both approaches use Lagrange polynomials as element shape functions at the Legendre–Gauss–Lobatto (LGL) points. The fluid and solid subdomains are coupled by applying the appropriate boundary conditions at the limiting interface. The proposed method is verified by means of two benchmark problems: wave propagation in an unbounded acoustic medium and the scattering of waves by an elastic inclusion. The convergence and the computational effort are evaluated for different h−p strategies. Numerical results show good agreement with the reference solution. Finally, the proposed method is used to study the pressure field generated by an array of elastic fluid-filled scatterers immersed in an acoustic medium.

An h–p version of the Chebyshev spectral collocation method for nonlinear delay differential equations

We develop and analyze a spectral collocation method based on the Chebyshev–Gauss–Lobatto points for nonlinear delay differential equations with vanishing delays. We derive an a priori error estimate in the H1‐norm that is completely explicit with respect to the local time steps and the local polynomial degrees. Several numerical examples are provided to illustrate the theoretical results.

The continuous Galerkin finite element methods for linear neutral delay differential equations

In the paper, the superconvergence of continuous Galerkin finite element methods (CGFEMs) for linear delay differential equations of neutral type is presented. By the orthogonal analysis method and under the suitable condition, it is proven that the finite element solution is superconvergent at the nodal points and Lobatto points. Numerical experiments further confirm the effectiveness and the superconvergence of the CGFEMs.

Fractional pseudospectral integration/differentiation matrix and fractional differential equations

In this paper, we present a new pseudospectral integration matrix which can be used to compute n−fold integrals of function f for any n∈R+. Also, it can be used to calculate the derivatives of f for any non-integer order α < 0. We use the Chebyshev interpolating polynomial for f at the Gauss–Lobatto points in [−1,1]. Less computational complexity and programming, much higher rate in running, calculating the integral/derivative of fractional order and its extraordinary accuracy, are advantages of this method in comparison with other known methods. We apply two approaches by using this matrix to solve some fractional differential equations with high accuracy. Some numerical examples are presented.

A New Direct Method for Solving Optimal Control Problem of Nonlinear Volterra–Fredholm Integral Equation via the Müntz–Legendre Polynomials

In this study, a new pseudo-spectral method is proposed to numerical solve of an optimal control problem whose constraint is given in terms of integral equation. The computational technique is based on a special form of the orthogonal Muntz–Legendre polynomials. Using the Legendre–Gauss–Lobatto points and Legendre–Gauss–Lobatto quadrature, the control and state functions are approximated by a finite combination of basis functions. Employing the method of this paper, the main optimal control problem becomes to an equivalent nonlinear programming problem. We also analyze the convergence of the proposed method by applying appropriate conditions. Finally, some examples are included to demonstrate the performance and accuracy of the introduced scheme.

Pricing multi-asset option problems: a Chebyshev pseudo-spectral method

The aim of this paper is to contribute a new second-order pseudo-spectral method via a non-uniform distribution of the computational nodes for solving multi-asset option pricing problems. In such problems, the prices are required to be as accurately as possible around the strike price. Accordingly, the proposed modification of the Chebyshev–Gauss–Lobatto points would concentrate on this area. This adaptation is also fruitful for the non-smooth payoffs which cause discontinuities in the strike price. The proposed scheme competes well with the existing methods such as finite difference, meshfree, and adaptive finite difference methods on several benchmark problems.

A composite pseudospectral method for solving multi-delay optimal control problems involving piecewise constant delay functions

An adaptive numerical method for solving multi-delay optimal control problems with piecewise constant delay functions is introduced. The proposed method is based on composite pseudospectral method using the well-known Legendre–Gauss–Lobatto points. In this approach, the main problem converts to a mathematical optimization problem whose solution is much more easier than the original one. The necessary conditions of optimality associated to nonlinear piecewise constant delay systems are derived. The method is easy to implement and provides very accurate results.

Numerical solution of Volterra–Fredholm integral equations using the collocation method based on a special form of the Müntz–Legendre polynomials

This paper presents a computational technique based on a special family of the Müntz–Legendre polynomials to solve a class of Volterra–Fredholm integral equations. The relationship between the Jacobi polynomials and Müntz–Legendre polynomials, in a particular state, are expressed. The proposed method reduces the integral equation into algebraic equations via the Chebyshev–Gauss–Lobatto points, so that the system matrix coefficients are obtained by the least squares approximation method. The useful properties of the Jacobi polynomials are exploited to analyze the approximation error. The performance and accuracy of our method are examined with some illustrative examples.

Stable evaluations of fractional derivative of the Müntz–Legendre polynomials and application to fractional differential equations

The aim of this paper is to present efficient and stable methods to compute Caputo fractional derivative (CFD) of the Müntz–Legendre polynomials based on three-term recurrence relations and Gauss–Jacobi quadrature rules. This approach with collocation method at Chebyshev–Gauss–Lobatto points has been applied for solving linear and nonlinear fractional multi-order differential equations (FDEs) described in Caputo sense. The main characteristic of spectral collocation method is that the problems reduce to linear or nonlinear systems of algebraic equations. In this work, for the first time, we present the new rates of convergence for projection error which are more accurate than the rate presented by Shen and Wang in Shen and Wang (2016). Moreover, we present convergence rate for spectral collocation method for linear FDEs with initial value on a finite interval and endpoint singularities. Also, we propose an error analysis for Jacobi–Gauss type quadrature and present a way to accelerate the convergence rate for singular integrands applied in this paper. Finally, the stability and applicability of the numerical approach and convergence analysis is demonstrated by some numerical examples.