Lobatto Points Research Articles

An approximate solution of the Blasius problem using spectral method

This paper aims at finding the numerical approximation of a classical Blasius flat plate problem using spectral collocation method. This technique is based on Chebyshev pseudospectral approach that involves the solution is approximated using Chebyshev polynomials, which are orthogonal polynomials defined on the interval [−1, 1]. The Chebyshev pseudospectral method employs Chebyshev- Gauss- Lobatto points, the extrema of the Chebyshev polynomials. The differential equation is approximated as a sum of Chebyshev polynomials. A differentiation matrix, based on these polynomials and their derivatives at the collocation points, transforms the differential equation into a system of algebraic equations. By evaluating the differential equation at these points and applying boundary conditions, the original boundary value problem reduced the solution to the solution of a system of algebraic equations. Solving for the coefficients of the polynomials yields the numerical approximation of the solution. The implementation of this method is carried out in Mathematica and its validity is ensured by comparing it with a built in MATLAB numerical routine called bvp4c.

Pseudospectral analysis for multidimensional fractional Burgers equation based on Caputo fractional derivative

This study presents the Chebyshev pseudospectral approach in time and space to approximate a solution to the time-fractional multidimensional Burgers equation. The suggested approach utilizes Chebyshev–Gauss–Lobatto (CGL) points in both spatial and temporal directions. To figure out the fractional derivative matrix at CGL points, we use the Caputo fractional derivative formula. Further, the Chebyshev fractional derivative matrix is utilized to reduce the given problem in an algebraic system of equations. The numerical approach known as the Newton–Raphson is implemented to get the desired results for the system. Error analysis for the set of values of ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \ u $$\\end{document} is done for various model examples of fractional Burgers equations, where ν\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ u $$\\end{document} represents the fractional order. The computed numerical results are in perfect agreement with the exact solutions.

Error estimates of the direct discontinuous Galerkin method for two-dimensional nonlinear convection–diffusion equations: Superconvergence analysis

This paper is concerned with superconvergence properties of the direct discontinuous Galerkin (DDG) method for two-dimensional nonlinear convection–diffusion equations. To copy with the complexity and difficulty caused by the nonlinear term, we utilize the method of Taylor expansion and correction idea to achieve our superconvergence goal. Specifically, we first adopt Taylor expansion to decompose the error into two parts: a high-order nonlinear term and a lower-order linear term. Then by using the idea of correction function for the linear term, a high order error bound is obtained. By doing so, a unified analysis of DDG method for general nonlinear problems is finally established. We prove that, for any piecewise tensor-product polynomials of degree k≥2, the DDG solution is superconvergent at nodes and Lobatto points, with an order of O(h2k) and O(hk+2), respectively. Moreover, superconvergence properties for the derivative approximation are also studied and the superconvergence points are identified at Gauss points, with an order of O(hk+1). Numerical experiments are presented to confirm the sharpness of all the theoretical findings.

A numerically robust and stable time–space pseudospectral approach for multidimensional generalized Burgers–Fisher equation

We study the pseudospectral discretization of a nonlinear multidimensional generalized Burgers–Fisher equation. The main objective of this study is to establish that the implementation of the pseudospectral method in time holds equal importance to its application in space when addressing nonlinearity. Typically, the literature recommends using a large number of grid points in the temporal direction to ensure satisfactory outcomes. However, the fact that we obtained excellent accuracy with a relatively small number of temporal grid points is a notable finding in this study. The Chebyshev–Gauss–Lobatto (CGL) points serve as the foundation for the recommended method. By applying the proposed method to the problem, a system of algebraic equations is obtained, which is subsequently solved using the Newton–Raphson technique. We have conducted stability analysis and error estimation for the proposed method. Different researchers’ considerations on test problems have been explored to illustrate the robustness and practicality of the approach presented. The results obtained using the proposed method demonstrate a high level of accuracy, surpassing the existing solutions by a significant margin.

Numerical behavior of the variable-order fractional Van der Pol oscillator

In this article, we investigate the behavior of a Van der Pol oscillator based on the variable-order Caputo fractional derivatives. After variable-order fractional modeling, we discretize the obtained equations using the Legendre–Gauss–Lobatto points and employ Lagrange interpolating functions. An algebraic system is gained that approximates the variables and their fractional derivatives. Also, an approach is suggested to calculate the differentiation matrix related to the variable-order Caputo fractional derivative. Moreover, an algorithm is presented for solving the variable-order Caputo fractional Van der Pol equation on large time-interval. Numerical simulations are provided to represent the applicability of the suggested method and to see the treatment of variable-order Caputo fractional Van der Pol oscillator.

High-Order Chebyshev Pseudospectral Tempered Fractional Operational Matrices and Tempered Fractional Differential Problems

This paper focuses on presenting an accurate, stable, efficient, and fast pseudospectral method to solve tempered fractional differential equations (TFDEs) in both spatial and temporal dimensions. We employ the Chebyshev interpolating polynomial for g at Gauss–Lobatto (GL) points in the range [−1,1] and any identically shifted range. The proposed method carries with it a recast of the TFDE into integration formulas to take advantage of the adaptation of the integral operators, hence avoiding the ill-conditioning and reduction in the convergence rate of integer differential operators. Via various tempered fractional differential applications, the present technique shows many advantages; for instance, spectral accuracy, a much higher rate of running, fewer computational hurdles and programming, calculating the tempered-derivative/integral of fractional order, and its spectral accuracy in comparison with other competitive numerical schemes. The study includes stability and convergence analyses and the elapsed times taken to construct the collocation matrices and obtain the numerical solutions, as well as a numerical examination of the produced condition number κ(A) of the resulting linear systems. The accuracy and efficiency of the proposed method are studied from the standpoint of the L2 and L∞-norms error and the fast rate of spectral convergence.

Efficient zigzag theory-based spectral element model for guided waves in composite structures containing delaminations

Lamb wave-based structural health monitoring offers excellent potential for real-time delamination detection in laminated structures. It necessitates fast and accurate numerical simulation of guided wave interaction with delaminations. Towards this objective, we present an efficient layerwise zigzag theory (ZIGT) based time-domain spectral element for wave propagation analysis of composite beam- and panel-type structures (strips) containing delamination at arbitrary locations. The ZIGT delivers accuracy like layerwise theories by allowing slope discontinuities in the in-plane displacement field at layer interfaces while maintaining computational efficiency like equivalent single-layer (ESL) theories, making it ideal for such analyses. The high-order elemental nodes at Lobatto points have only four displacement variables, irrespective of the number of layers in the laminate. Delamination is modelled using the region approach, adapting the recently proposed hybrid point-least squares continuity method to satisfy the nonlinear displacement field continuity at the delamination fronts. The model’s efficacy is examined vis-à-vis elasticity-based elements, ESL theory-based elements, and the ZIGT-based standard finite element for free vibration and guided wave propagation responses of composite strips featuring multiple delaminations. The present model shows superior overall efficiency, accuracy, and convergence. An application of the model is also explored for identifying delamination locations from Lamb wave velocity fields.

An operator splitting Legendre-tau spectral method for Maxwell’s equations with nonlinear conductivity in two dimensions

In the paper, an operator splitting Legendre-tau spectral method is proposed to solve Maxwell’s equations with nonlinear conductivity in two dimensions. By the implicit–explicit numerical scheme, the linear and nonlinear parts are treated in different ways. The linear terms are treated by the Legendre-tau spectral method implicitly, and the nonlinear terms are treated by some collocation methods explicitly. The scheme uses polynomial spaces of different degrees to approximate the electric and magnetic fields, respectively, so that they can be decoupled in computation. The splitting leap frog Crank–Nicolson method is applied in time discretization, which is a three-level scheme with the nonlinear term being treated explicitly in intermediate level. With the application of inverse inequality, for the Legendre interpolation operator, optimal L2-error estimates are achieved for the proposed scheme. In numerical examples, the nonlinear terms are computed at Chebyshev–Gauss–Lobatto points by the fast Legendre transform. Numerical results validate the efficiency and spectral accuracy of the scheme.

CHEBYSHEV PSEUDOSPECTRAL METHOD FOR DUFFING NONLINEAR DIFFERENTIAL EQUATIONS

The system of Duffing nonlinear differential equations is often used in dynamics, which are known to describe many important oscillating phenomena in nonlinear engineering systems. This article presents the pseudospectral method to calculate numerical solutions for nonlinear Duffing differential equations on the interval [–1, 1]. This method is based on the differentiation matrix using the Chebyshev Gauss – Lobatto points. To find numerical solutions of the nonlinear Duffing differential equations, we have built an iterative procedure. The software used for the calculations in this study was Mathematica 10.4. The numerical results of the comparison show that this solution had a high degree of accuracy and very small errors.

Multi-domain multivariate spectral collocation method for (2+1) dimensional nonlinear partial differential equations

The novelty of this work rests upon the use of the domain partitioning technique in time variable when discretizing the domain of solution in spectral collocation algorithm. The single domain multivariate spectral collocation based methods have been proven to be effective in solving time-dependent partial differential equations (PDEs) defined over small time domains. However, there is a significant loss of accuracy as time computational domain proliferates and also when the number of grid nodes approaches a definite particular number. Therefore, the establishment of the new innovative multi-domain multivariate spectral quasilinearisation method (MDMV-SQLM) is described for the purpose of solving (2+1) dimensional nonlinear PDEs defined on large time intervals. The main output of this study is confirmation that minimizing the size of time computational domain at each subinterval assures sufficiently accurate results that are attained using minimal number of nodal points and less computational time. The solution algorithm involves partitioning the time domain into multiple non-overlapping sub-domains, simplification of the nonlinear PDEs using the quasilinearisation method and assumption of approximate solutions using triple Lagrange interpolating polynomials with Chebyshev–Gauss–Lobatto (CGL) points. The multi-domain spectral collocation procedure is executed on the linear systems of algebraic equations, where the subsequent matrix systems are solved separately in every time sub-interval with the continuity equation essentially used in obtaining initial conditions in the next subintervals. MATLAB software is used to implement the solution algorithm and numerical results are demonstrated graphically and in tabular form. To highlight the efficaciousness and accuracy of the MDMV-SQLM, error estimates, condition numbers and computational time are presented for well known (2+1) dimensional nonlinear initial-Dirichlet boundary value problems. The adoption of the domain decomposition technique is efficacious in suppressing the numerical challenges linked to large matrices and ill-conditioned nature of the resulting coefficient matrix. Also, the communicated results confirm that the numerical scheme is computationally cheap, fast and yield extremely accurate and stable results with the aid of fewer number of grid points for large time domains.

Solving two-dimensional bioheat optimal control problem in solid and vessel domain by pseudospectral discretization

In this paper, an optimal control problem subject to an energy transfer equation in thermally significant blood vessel coupled with the Pennes governing equation is solved. The proposed dynamical system is modeled by a system of conduction-reaction and convection-conduction equations. Chebyshev–Gauss–Lobatto (CGL) and Legendre–Gauss–Lobatto (LGL) points are applied to discretize optimal control problem. Then, the problem transferred into a quadratic programming problem using domain decomposition and matching the solution and its first derivative across the related interfaces. The Tikhonov regularization method is used to compute the optimal regularization parameter for the objective function. Some theories have been discussed to prove that the discrete problem has a solution and it is also consistent with the continuous problem. Numerical results are compared and numerical convergent is shown. Optimal and non-optimal external heat source is compared and it is shown that the mathematical modeling is effective in saving the external heat energy. Numerical results show that by this process of heating at the final time, the temperature is very close to the desired temperature without damaging the vessel domain. Numerical results and corresponding graphs are depicted.

An entropy–stable p–adaptive nodal discontinuous Galerkin for the coupled Navier–Stokes/Cahn–Hilliard system

We develop a novel entropy–stable discontinuous Galerkin approximation of the incompressible Navier–Stokes/Cahn–Hilliard system for p–non–conforming elements. This work constitutes an evolution of the work presented by Manzanero et al. ((2020) [10]), as it extends the discrete analysis into supporting p–adaptation (p–refinement/coarsening). The scheme is based on the summation–by–parts simultaneous–approximation term property along with Gauss–Lobatto points and suitable numerical fluxes. The p–non–conforming elements are connected through the classic mortar method, the use of central fluxes for the inviscid terms, and the BR1 scheme with additional dissipation for the viscous fluxes. The scheme is proven to retain its properties of the original conforming scheme when transitioning to p–non–conforming elements and to mimic the continuous entropy analysis of the model. We focus on dynamic polynomial adaptation as the applications of interest are unsteady multiphase flows. In this work, we introduce a heuristic adaptation criterion that depends on the location of the interface between the different phases and utilises the convection velocity to predict the movement of the interface. The scheme is verified to be total phase conserving, entropy–stable and freestream preserving for curvilinear p–non–conforming meshes. We also present the results for a rising bubble simulation and we show that for the same accuracy we get a ×2 to ×6 reduction in the degrees of freedom and a 41% reduction in the computational time. We compare our results for the three–dimensional dam break test case against experimental and numerical data and we show that a ×4.3 to ×9.5 reduction of the degrees of freedom and a 51% reduction in the computational time can be achieved compared to the p–uniform solution.

Collocation approaches to the mathematical model of an Euler–Bernoulli beam vibrations

Taylor-Matrix and Hermite-Matrix Collocation methods are presented to obtain the modal vibration behavior of an Euler–Bernoulli beam. The methods provide approximate solutions in the truncated Taylor series and the truncated Hermite series by using Chebyshev–Lobatto collocation points and operational matrices. The approaches are applied to the well-known transverse vibration model of a simply-supported Euler–Bernoulli type beam. The beam is assumed under the effect of external harmonic force with spatially varying amplitude. Firstly, the model problem is transformed into a system of linear algebraic equations. Then the approximate solution is computed by solving the obtained algebraic system. The proposed methods are compared with the exact solutions. Mode shapes of the first three fundamental frequencies are obtained for each approach. The convergence behaviors of transverse displacements and absolute errors are calculated for each mode. The numerical results are shown and compared. Based on the given figures and tables, one can state that the methods are remarkable, dependable, and accurate for approaching the mentioned model problems.

Approximation Properties of Chebyshev Polynomials in the Legendre Norm

In this paper, we present some important approximation properties of Chebyshev polynomials in the Legendre norm. We mainly discuss the Chebyshev interpolation operator at the Chebyshev–Gauss–Lobatto points. The cases of single domain and multidomain for both one dimension and multi-dimensions are considered, respectively. The approximation results in Legendre norm rather than in the Chebyshev weighted norm are given, which play a fundamental role in numerical analysis of the Legendre–Chebyshev spectral method. These results are also useful in Clenshaw–Curtis quadrature which is based on sampling the integrand at Chebyshev points.

On [formula omitted]-Chebyshev functions and points of the interval

In this paper, we introduce the class of (β,γ)-Chebyshev functions and corresponding points, which can be seen as a family of generalized Chebyshev polynomials and points. For the (β,γ)-Chebyshev functions, we prove that they are orthogonal in certain subintervals of [−1,1] with respect to a weighted arc-cosine measure. In particular we investigate the cases where they become polynomials, deriving new results concerning classical Chebyshev polynomials of first kind. Besides, we show that subsets of Chebyshev and Chebyshev–Lobatto points are instances of (β,γ)-Chebyshev points. We also study the behavior of the Lebesgue constants of the polynomial interpolant at these points on varying the parameters β and γ.

A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

A novel free–energy stable discontinuous Galerkin method is developed for the Cahn–Hilliard equation with non–conforming elements. This work focuses on dynamic polynomial adaptation (p–refinement) and constitutes an extension of the method developed by Manzanero et al. (2020) [8], which makes use of the summation–by–parts simultaneous–approximation term technique along with Gauss–Lobatto points and the Bassi–Rebay 1 (BR1) scheme. The BR1 numerical flux accommodates non–conforming elements, which are connected through the mortar method. The scheme has been analytically proven to retain its free–energy stability when transitioning to non-conforming elements. Furthermore, a methodology to perform the adaptation is introduced based on the knowledge of the location of the interface between phases. The adaptation methodology is tested for its accuracy and effectiveness through a series of steady and unsteady test cases. We test the scheme for freestream preservation and primary quantity conservation on non–conforming curvilinear meshes. We solve a steady one–dimensional interface test case to initially examine the accuracy of the adaptation. Furthermore, we study the formation of a static bubble in two dimensions and verify that the accuracy of the solver is maintained while the degrees of freedom decrease to less than half compared to the uniform solution. Lastly, we examine an unsteady case such as the spinodal decomposition and show that the same results for the free–energy are recovered, with a 35% reduction of the degrees of freedom for the two–dimensional case considered and a 48% reduction for the three–dimensional case.

Pseudospectral analysis and approximation of two‐dimensional fractional cable equation

In recent years, major attention has been devoted to fractional‐order partial differential equations since they seem to be more efficient in modeling complex processes. Fractional‐order partial differential equations are considered as generalizations of classical integer‐order partial differential equations. In many areas of electrophysiology and in modeling neuronal dynamics, the cable equation plays a major role. In this article, the authors constructed to approximate the numerical solutions of the two‐dimensional fractional cable equation using the time‐space Jacobi pseudospectral method. The proposed method is established in both time and space to approximate the solutions. Moreover, we use fractional Lagrange interpolants polynomial as a test function, which satisfies the Kronecker delta property at Jacobi–Gauss–Lobatto (JGL) points. The fractional derivative is defined in the modified Riemann–Liouville fractional derivatives formula at JGL points. Using the proposed method, the approximate solution is obtained by solving a diagonally block system of nonlinear algebraic equations. The stability analysis of the proposed method, uniqueness of the solution, and error estimate are also provided. Finally, numerical solutions are demonstrated to justify the theoretical results and confirm the expected convergence rate. The pseudospectral solutions are more accurate as compared to the available results to date in the same vicinity.

Efficient operator splitting and spectral methods for the time-space fractional Black–Scholes equation

In this paper, we aim at developing improved L1 operator splitting method and spectral method for Black–Scholes differential systems with fractional derivatives in both time and space. We address the challenge of slow convergence in time and provide efficient approaches to handle complementarity conditions in the differential system. We design an operator splitting method to decouple the constraints together with an improved L1 approximation to accommodate nonsmooth initial data. In the space dimension, a spectral collocation method based on Legendre–Gauss–Lobatto points was employed. To show the efficiency and accuracy, we benchmark our improved L1 method with the original L1 method within the operator splitting framework. For European options, both the L1 method and improved L1 method provide the convergence order of 2−γ, where γ is fractional order in time. For American options, it is necessary to apply the improved L1 method to achieve the order of 2−γ.

Numerical and theoretical discussions for solving nonlinear generalized Benjamin–Bona–Mahony–Burgers equation based on the Legendre spectral element method

AbstractThis article is devoted to solving numerically the nonlinear generalized Benjamin–Bona–Mahony–Burgers (GBBMB) equation that has several applications in physics and applied sciences. First, the time derivative is approximated by using a finite difference formula. Afterward, the stability and convergence analyses of the obtained time semi‐discrete are proven by applying the energy method. Also, it has been demonstrated that the convergence order in the temporal direction isO(dt). Second, a fully discrete formula is acquired by approximating the spatial derivatives via Legendre spectral element method. This method uses Lagrange polynomial based on Gauss–Legendre–Lobatto points. An error estimation is also given in detail for full discretization scheme. Ultimately, the GBBMB equation in the one‐ and two‐dimension is solved by using the proposed method. Also, the calculated solutions are compared with theoretical solutions and results obtained from other techniques in the literature. The accuracy and efficiency of the mentioned procedure are revealed by numerical samples.

An MHD Stokes eigenvalue problem and its approximation by a spectral collocation method

An eigenvalue problem is introduced for the magnetohydrodynamic (MHD) Stokes equations describing the flow of a viscous and electrically conducting fluid in a duct under the influence of a uniform magnetic field. The solution of the eigenproblem is approximated by using a spectral collocation method that is based on vanishing the residual equation at the collocation points on the physical domain which are chosen to be the Chebyshev–Gauss–Lobatto points. As the solutions are sought in the physical space, the approximations to the derivatives of the unknowns are directly evaluated. The equations are formulated in the primitive variables, and hence with inclusion of the continuity equation, the discretization of the operator results in a generalized eigenproblem with zero diagonal entries. Therefore, a penalty method is applied to circumvent the degeneracy where a perturbed form of the problem is considered, and a zero mean pressure value is introduced. The numerical prospects of the algorithm are investigated and demonstrated by a number of characteristic tests. The key features of interest are the effects of introducing a magnetic field on the eigenspectrum focusing mainly on the change of the fundamental eigenpairs, and the consequential variation of the eigenstructure with the magnetic field. The mechanisms that underlie these effects are examined by the numerical model proposed, the implications of these effects are presented, and it is shown that the flow field is considerably affected with the introduction of a magnetic field into the physical model.