Gauss-Lobatto Quadrature Rules Research Articles

A fully spectral collocation approximation for multi-dimensional fractional Schrödinger equations

A shifted Legendre collocation method in two consecutive steps is developed and analyzed to numerically solve one- and two-dimensional time fractional Schrödinger equations (TFSEs) subject to initial-boundary and non-local conditions. The first step depends mainly on shifted Legendre Gauss–Lobatto collocation (SL-GL-C) method for spatial discretization; an expansion in a series of shifted Legendre polynomials for the approximate solution and its spatial derivatives occurring in the TFSE is investigated. In addition, the Legendre–Gauss–Lobatto quadrature rule is established to treat the nonlocal conservation conditions. Thereby, the expansion coefficients are then determined by reducing the TFSE with its nonlocal conditions to a system of fractional differential equations (SFDEs) for these coefficients. The second step is to propose a shifted Legendre Gauss–Radau collocation (SL-GR-C) scheme, for temporal discretization, to reduce such system into a system of algebraic equations which is far easier to be solved. The proposed collocation scheme, both in temporal and spatial discretizations, is successfully extended to solve the two-dimensional TFSE. Numerical results are carried out to confirm the spectral accuracy and efficiency of the proposed algorithms. By selecting relatively limited Legendre Gauss–Lobatto and Gauss–Radau collocation nodes, we are able to get very accurate approximations, demonstrating the utility and high accuracy of the new approach over other numerical methods.

An efficient collocation algorithm for multidimensional wave type equations with nonlocal conservation conditions

In this paper, we derive and analyze an efficient spectral collocation algorithm to solve numerically some wave equations subject to initial-boundary nonlocal conservation conditions in one and two space dimensions. The Legendre pseudospectral approximation is investigated for spatial approximation of the wave equations. The Legendre–Gauss–Lobatto quadrature rule is established to treat the nonlocal conservation conditions, and then the problem with its nonlocal conservation conditions are reduced to a system of ODEs in time. As a theoretical result, we study the convergence of the solution for the one-dimensional case. In addition, the proposed method is extended successfully to the two-dimensional case. Several numerical examples with comparisons are given. The computational results indicate that the proposed method is more accurate than finite difference method, the method of lines and spline collocation approach.

An efficient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system

In this paper, we derive an efficient spectral collocation algorithm to solve numerically the nonlinear complex generalized Zakharov system (GZS) subject to initial-boundary conditions. The Jacobi pseudospectral approximation is investigated for spatial approximation of the GZS. It possesses the spectral accuracy in space. The Jacobi–Gauss–Lobatto quadrature rule is established to treat the boundary conditions, and then the problem with its boundary conditions is reduced to a system of ordinary differential equations in time variable. This scheme has the advantage of allowing us to obtain the spectral solution in terms of the Jacobi parameters α and β, which therefore means that the algorithm holds a number of collocation methods as special cases. Finally, two illustrative examples are implemented to assess the efficiency and high accuracy of the Jacobi pseudo-spectral scheme.

Pseudospectral methods based on nonclassical orthogonal polynomials for solving nonlinear variational problems

Two direct pseudospectral methods based on nonclassical orthogonal polynomials are proposed for solving finite-horizon and infinite-horizon variational problems. In the proposed finite-horizon and infinite-horizon methods, the rate variables are approximated by the Nth degree weighted interpolant, using nonclassical Gauss-Lobatto and Gauss points, respectively. Exponential Freud type weights are introduced for both of nonclassical orthogonal polynomials and weighted interpolation. It is shown that the absolute error in weighted interpolation is dependent on the selected weight, and the weight function can be tuned to improve the quality of the approximation. In the finite-horizon scheme, the functional is approximated based on Gauss-Lobatto quadrature rule, thereby reducing the problem to a nonlinear programming one. For infinite-horizon problems, an strictly monotonic transformation is used to map the infinite domain onto a finite interval. We transcribe the transformed problem to a nonlinear programming using Gauss quadrature rule. Numerical examples demonstrate the accuracy of the proposed methods.

Erratum to: On Gauss-type quadrature formulas with prescribed nodes anywhere on the real line

This is an erratum to our previous paper in this journal. In the paper we investigated among other things the existence of Gauss-type quadrature formulas with two prescribed nodes and a maximal domain of exactness. At some instances we claimed that the weights in the quadrature formulas are positive, while this only holds under more restrictive conditions than given. These are the corrections that have to be implemented. 1. In the discussion before Theorem 2.15, the case GL3c should be completed with the following additional information. T (b) = fn(b) iff a is a node of the quadrature and T (a) = fn(a) iff b is a node. If all the nodes are in (a, b) the corresponding weights will be positive. If a or b is one of the nodes, then the corresponding weight cannot be guaranteed to be positive. 2. The last sentence of Theorem 2.15 should be changed as follows. In all the cases where the Gauss-Lobatto quadrature rule exist, it must be of interpolatory type and the positivity of the weights for all the nodes inside (a, b) is guaranteed. The positivity of the weight corresponding to a node in one of the endpoints of the interval can not be guaranteed. 3. In Corollary 2.17, the sentence “The weights of the quadrature are positive” should be replaced by the following: The weights corresponding to the nodes different from a if μ(x) = (x− a)μ(x) (respectively different from b if μ(x) = (b− x)μ(x)) are positive, but the weight for a (respectively b) can not be guaranteed to be positive. More details about conditions that guarantee that all the weights are positive are provided in [1].

A DIFFERENTIAL QUADRATURE FINITE ELEMENT METHOD

This paper studies the differential quadrature finite element method (DQFEM) systematically, as a combination of differential quadrature method (DQM) and standard finite element method (FEM), and formulates one- to three-dimensional (1-D to 3-D) element matrices of DQFEM. It is shown that the mass matrices of C 0 finite element in DQFEM are diagonal, which can reduce the computational cost for dynamic problems. The Lagrange polynomials are used as the trial functions for both C 0 and C 1 differential quadrature finite elements (DQFE) with regular and/or irregular shapes, this unifies the selection of trial functions of FEM. The DQFE matrices are simply computed by algebraic operations of the given weighting coefficient matrices of the differential quadrature (DQ) rules and Gauss-Lobatto quadrature rules, which greatly simplifies the constructions of higher order finite elements. The inter-element compatibility requirements for problems with C 1 continuity are implemented through modifying the nodal parameters using DQ rules. The reformulated DQ rules for curvilinear quadrilateral domain and its implementation are also presented due to the requirements of application. Numerical comparison studies of 2-D and 3-D static and dynamic problems demonstrate the high accuracy and rapid convergence of the DQFEM.

High‐accuracy differential quadrature finite element method and its application to free vibrations of thin plate with curvilinear domain

AbstractBased on the differential quadrature (DQ) rule, the Gauss Lobatto quadrature rule and the variational principle, a DQ finite element method (DQFEM) is proposed for the free vibration analysis of thin plates. The DQFEM is a highly accurate and rapidly converging approach, and is distinct from the differential quadrature element method (DQEM) and the quadrature element method (QEM) by employing the function values themselves in the trial function for the title problem.The DQFEM, without using shape functions, essentially combines the high accuracy of the differential quadrature method (DQM) with the generality of the standard finite element formulation, and has superior accuracy to the standard FEM and FDM, and superior efficiency to the p‐version FEM and QEM in calculating the stiffness and mass matrices.By incorporating the reformulated DQ rules for general curvilinear quadrilaterals domains into the DQFEM, a curvilinear quadrilateral DQ finite plate element is also proposed. The inter‐element compatibility conditions as well as multiple boundary conditions can be implemented, simply and conveniently as in FEM, through modifying the nodal parameters when required at boundary grid points using the DQ rules. Thus, the DQFEM is capable of constructing curvilinear quadrilateral elements with any degree of freedom and any order of inter‐element compatibilities. A series of frequency comparisons of thin isotropic plates with irregular and regular planforms validate the performance of the DQFEM. Copyright © 2009 John Wiley & Sons, Ltd.

Time-harmonic elasticity with controllability and higher-order discretization methods

The time-harmonic solution of the linear elastic wave equation is needed for a variety of applications. The typical procedure for solving the time-harmonic elastic wave equation leads to difficulties solving large-scale indefinite linear systems. To avoid these difficulties, we consider the original time dependent equation with a method based on an exact controllability formulation. The main idea of this approach is to find initial conditions such that after one time-period, the solution and its time derivative coincide with the initial conditions. The wave equation is discretized in the space domain with spectral elements. The degrees of freedom associated with the basis functions are situated at the Gauss–Lobatto quadrature points of the elements, and the Gauss–Lobatto quadrature rule is used so that the mass matrix becomes diagonal. This method is combined with the second-order central finite difference or the fourth-order Runge–Kutta time discretization. As a consequence of these choices, only matrix–vector products are needed in time dependent simulation. This makes the controllability method computationally efficient.

Szegő–Lobatto quadrature rules

Gauss-type quadrature rules with one or two prescribed nodes are well known and are commonly referred to as Gauss–Radau and Gauss–Lobatto quadrature rules, respectively. Efficient algorithms are available for their computation. Szegő quadrature rules are analogs of Gauss quadrature rules for the integration of periodic functions; they integrate exactly trigonometric polynomials of as high degree as possible. Szegő quadrature rules have a free parameter, which can be used to prescribe one node. This paper discusses an analog of Gauss–Lobatto rules, i.e., Szegő quadrature rules with two prescribed nodes. We refer to these rules as Szegő–Lobatto rules. Their properties as well as numerical methods for their computation are discussed.

On numerical improvement of Gauss–Lobatto quadrature rules

It is well known that Gauss–Lobatto quadrature rule ∫ - 1 1 f ( x ) d x ≃ ∑ i = 1 n w i f ( x i ) + pf ( - 1 ) + qf ( 1 ) , is exact for polynomials of degree at most 2 n + 1. In this paper we are going to find a formula which is approximately exact for monomials x j , j = 0, 1, …, 2 n + 3 instead of being analytically exact for monomials x j , j = 0, 1, …, 2 n + 1. We also consider a class of functions for which the new formula produces better results.

Discrete variable representation for singular Hamiltonians

We discuss the application of the discrete variable representation (DVR) to Schrödinger problems which involve singular Hamiltonians. Unlike recent authors who invoke transformations to rid the eigenvalue equation of singularities at the cost of added complexity, we show that an approach based solely on an orthogonal polynomial basis is adequate, provided the Gauss-Lobatto or Gauss-Radau quadrature rule is used. This ensures that the mesh contains the singular points and by simply discarding the DVR functions corresponding to those points, all matrix elements become well behaved, the boundary conditions are satisfied, and the calculation is rapidly convergent. The accuracy of the method is demonstrated by applying it to the hydrogen atom. We emphasize that the method is equally capable of describing bound states and continuum solutions.

Symmetric Gauss–Lobatto and Modified Anti-Gauss Rules

The present paper is concerned with symmetric Gauss–Lobatto quadrature rules, i.e., with Gauss–Lobatto rules associated with a nonnegative symmetric measure on the real axis. We propose a modification of the anti-Gauss quadrature rules recently introduced by Laurie, and show that the symmetric Gauss–Lobatto rules are modified anti-Gauss rules. It follows that for many integrands, symmetric Gauss quadrature rules and symmetric Gauss–Lobatto rules give quadrature errors of opposite sign.

Computation of Gauss-Kronrod quadrature rules

Recently Laurie presented a new algorithm for the computation of (2n+1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. This algorithm first determines a symmetric tridiagonal matrix of order 2n + 1 from certain mixed moments, and then computes a partial spectral factorization. We describe a new algorithm that does not require the entries of the tridiagonal matrix to be determined, and thereby avoids computations that can be sensitive to perturbations. Our algorithm uses the consolidation phase of a divide-and-conquer algorithm for the symmetric tridiagonal eigenproblem. We also discuss how the algorithm can be applied to compute Kronrod extensions of Gauss-Radau and Gauss-Lobatto quadrature rules. Throughout the paper we emphasize how the structure of the algorithm makes efficient implementation on parallel computers possible. Numerical examples illustrate the performance of the algorithm.

High-order Gauss-Lobatto formulae

Currently, the method of choice for computing the (n+2)-point Gauss–Lobatto quadrature rule for any measure of integration is to first generate the Jacobi matrix of order n+2 for the measure at hand, then modify the three elements at the right lower corner of the matrix in a manner proposed in 1973 by Golub, and finally compute the eigenvalues and first components of the respective eigenvectors to produce the nodes and weights of the quadrature rule. In general, this works quite well, but when n becomes large, underflow problems cause the method to fail, at least in the software implementation provided by us in 1994. The reason is the singularity (caused by underflow) of the 2×2 system of linear equations that is used to compute the modified matrix elements. It is shown here that in the case of arbitrary Jacobi measures, these elements can be computed directly, without solving a linear system, thus allowing the method to function for essentially unrestricted values of n. In addition, it is shown that all weights of the quadrature rule can also be computed explicitly, which not only obviates the need to compute eigenvectors, but also provides better accuracy. Numerical comparisons are made to illustrate the effectiveness of this new implementation of the method.

The remainder term for analytic functions of Gauss-Lobatto quadratures

For analytic functions the remainder term of Gauss-Lobatto quadrature rules can be represented as a contour integral with a complex kernel. In this paper the kernel is studied on elliptic contours for the Chebyshev weight functions of the second, third, and fourth kind. Starting from explicit expressions of the corresponding kernels the location of their maximum modulus on ellipses is determined. This gives an answer to Gautschi's (1991) conjectures on the location of the maximum point for these kernels. Finally, some extensions to Gaussian quadrature rules as well as numerical examples are given.

Direct optimization using collocation based on high-order Gauss-Lobatto quadrature rules

The method of collocation and nonlinear programming has been used recently to solve a number of optimal control problems. In this method polynomials are commonly used to represent the state variable time histories over subintervals of the total time of interest. These polynomials correspond to a family of modified-Gaussian quadrature rules known as the Gauss-Lobatto rules. Presently, relatively low-order rules from the Gauss-Lobatto family, such as the trapezoid and Simpson's rule, are used to construct collocation solution schemes. In this work higher-order Gauss-Lobatto quadrature rules are formulated using collocation point selection based on a particular family of Jacob! polynomials. The advantage of using a quadrature rule of higher order is that the approximation using the higher degree polynomial may be more accurate, due to finite precision arithmetic, than a formulation based on a lower degree polynomial. In addition, the number of subintervals and, therefore, the number of nonlinear programming parameters needed to solve a problem accurately may be significantly reduced from that required if the conventional trapezoidal or Simpson's quadrature schemes are used. An optimal trajectory maximizing final energy for a low-thrust spacecraft is used to demonstrate the benefits of using the higher-order schemes.

The pseudospectral Legendre method for discretizing optimal control problems

This paper presents a computational technique for optimal control problems including state and control inequality constraints. The technique is based on spectral collocation methods used in the solution of differential equations. The system dynamics are collocated at Legendre-Gauss-Lobatto points. The derivative x/spl dot/(t) of the state x(t) is approximated by the analytic derivative of the corresponding interpolating polynomial. State and control inequality constraints are collocated at Legendre-Gauss-Lobatto nodes. The integral involved in the definition of the performance index is discretized based on the Gauss-Lobatto quadrature rule. The optimal control problem is thereby converted into a mathematical programming program. Thus existing, well-developed optimization algorithms may be used to solve the transformed problem. The method is easy to implement, capable of handling various types of constraints, and yields very accurate results. Illustrative examples are included to demonstrate the capability of the proposed method, and a comparison is made with existing methods in the literature.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Higher-order finite elements with mass-lumping for the 1D wave equation

This paper is devoted to the construction and analysis of a method, higher order in space and time, for solving the one-dimensional wave equation. This method is based on P 3 Lagrange finite elements with mass-lumping which avoids the inversion of a mass matrix at each time-step. The mass-lumping implies to make the abscissae of the interior points coincide with these of the Gauss-Lobatto quadrature rule. A Fourier analysis of the method for a regular mesh points out a superconvergence result. The gain of accuracy is illustrated by numerical experiments.

Gauss—Radau and Gauss—Lobatto quadratures with double end points

We develop explicit formulae for generalized Gauss—Radau and Gauss—Lobatto quadrature rules having end points of multiplicity 2 and containing Chebyshev weight functions of any of the four kinds.

The remainder term for analytic functions of Gauss-Radau and Gauss-Lobatto quadrature rules with multiple end points

A study is undertaken of the kernels in the contour integral representations of the remainder terms for Gauss-Radau and Gauss-Lobatto quadrature rules over the interval [−1, 1]. It is assumed that the respective end points in these rules have multiplicity two, and that integration is with respect to one of the four Chebyshev weight functions. Of particular interest is the location on the contour where the modulus of the kernel attains its maximum value. Only elliptic contours are considered having foci at the points ± 1.