Gauss-Radau Quadrature Research Articles

2-point left Radau-type inequalities via s-convexity

Abstract Convexity is a fundamental concept in analysis. Over the past few decades, many significant error bounds have been established for various quadrature rules using different types of convexity. This paper focuses on the Gauss–Radau quadrature formula. Initially, we introduce a novel identity related to 2-point left Radau-type rule. Next, we derive several integral inequalities for functions whose first derivatives are s-convex in the second sense. Finally, we present applications to special means to demonstrate the effectiveness of our results.

Applications of Fractional Differentiation Matrices in Solving Caputo Fractional Differential Equations

This paper pursues obtaining Jacobi spectral collocation methods to solve Caputo fractional differential equations numerically. We used the shifted Jacobi–Gauss–Lobatto or Jacobi–Gauss–Radau quadrature nodes as the collocation points and derived the fractional differentiation matrices for Caputo fractional derivatives. With the fractional differentiation matrices, the fractional differential equations were transformed into linear systems, which are easier to solve. Two types of fractional differential equations were used for the numerical simulations, and the numerical results demonstrated the fast convergence and high accuracy of the proposed methods.

A SPACE-TIME SPECTRAL COLLOCATION METHOD FOR SOLVING THE VARIABLE-ORDER FRACTIONAL FOKKER-PLANCK EQUATION

A numerical approach for solving the variable-order fractional FokkerPlanck equation (VO-FFPE) is proposed. The computational scheme is based on the shifted Legendre Gauss-Lobatto and the shifted Chebyshev GaussRadau collocation methods. The VO-FFPE is written as a truncated series of shifted Legendre and shifted Chebyshev polynomials for space and time variables, respectively. The residuals of the VO-FFPE at the shifted Legendre Gauss-Lobatto and shifted Chebyshev Gauss-Radau quadrature points are estimated. The original problem is converted into a system of algebraic equations that can be solved easily. Several examples are presented to demonstrate the efficacy of the technique.

Error profile for discontinuous Galerkin time stepping of parabolic PDEs

We consider the time discretization of a linear parabolic problem by the discontinuous Galerkin (DG) method using piecewise polynomials of degree at most r − 1 in t, for r ≥ 1 and with maximum step size k. It is well known that the spatial L2-norm of the DG error is of optimal order kr globally in time, and is, for r ≥ 2, superconvergent of order k2r− 1 at the nodes. We show that on the n th subinterval (tn− 1,tn), the dominant term in the DG error is proportional to the local right Radau polynomial of degree r. This error profile implies that the DG error is of order kr+ 1 at the right-hand Gauss–Radau quadrature points in each interval. We show that the norm of the jump in the DG solution at the left end point tn− 1 provides an accurate a posteriori estimate for the maximum error over the subinterval (tn− 1,tn). Furthermore, a simple post-processing step yields a continuous piecewise polynomial of degree r with the optimal global convergence rate of order kr+ 1. We illustrate these results with some numerical experiments.

A new stable numerical method for Mellin integral equations in weighted spaces with uniform norm

In this paper a new modified Nystrom method is proposed to solve linear integral equations of the second kind with fixed singularities of Mellin convolution type. It is based on the Gauss–Radau quadrature formula with a suitable Jacobi weight. The stability and convergence of the method is proved in weighted spaces with uniform norm. Moreover, an error estimate of the numerical solution is given under certain assumptions on the Mellin kernel. The efficiency of the method is shown through some examples. The numerical results also confirm that the error estimate is sharp.

Generalized block anti-Gauss quadrature rules

Golub and Meurant describe how pairs of Gauss and Gauss–Radau quadrature rules can be applied to determine inexpensively computable upper and lower bounds for certain real-valued matrix functionals defined by a symmetric matrix. However, there are many matrix functionals for which their technique is not guaranteed to furnish upper and lower bounds. In this situation, it may be possible to determine upper and lower bounds by evaluating pairs of Gauss and anti-Gauss rules. Unfortunately, it is difficult to ascertain whether the values determined by Gauss and anti-Gauss rules bracket the value of the given real-valued matrix functional. Therefore, generalizations of anti-Gauss rules have recently been described, such that pairs of Gauss and generalized anti-Gauss rules may determine upper and lower bounds for real-valued matrix functionals also when pairs of Gauss and (standard) anti-Gauss rules do not. The available generalization requires the matrix that defines the functional to be real and symmetric. The present paper reviews available anti-Gauss and generalized anti-Gauss rules and extends them in several ways that allow applications in new situations. In particular, the genarlized anti-Gauss rules for a real-valued non-negative measure described in Pranic and Reichel (J Comput Appl Math 284:235–243, 2015) are extended to allow the estimation of the error in matrix functionals defined by a non-symmetric matrix, as well as to matrix-valued matrix functions. Modifications that give simpler formulas and thereby make the application of the rules both easier and applicable to a larger class of problems also are described.

A numerical method for the solution of integral equations of Mellin type

We are interested in the numerical solution of second kind integral equations of Mellin convolution type. We describe a modified Nyström method based on the Gauss–Lobatto or Gauss–Radau quadrature rule. Under certain assumptions on the Mellin kernel, we prove the stability and the convergence of the proposed procedure and also derive error estimates. Finally, some test problems are solved and the numerical results showing the effectiveness of our method are presented.

The remainder term of Gauss-Radau quadrature rule with single and double end point

The remainder term of quadrature formula can be represented as a contour integral with a complex kernel. We study the kernel on elliptic contours for Gauss?Radau quadrature formula with the Chebyshev weight function of the second kind with double and single end point. Starting from the explicit expression of the corresponding kernel, derived by Gautschi and Li, we determine the locations on the ellipses where the maximum modulus of the kernel is attained.

Numerical solution of initial-boundary system of nonlinear hyperbolic equations

In this article, we present a numerical approximation of the initial-boundary system of nonlinear hyperbolic equations based on spectral Jacobi-Gauss-Radau collocation (J-GR-C) method. A J-GR-C method in combination with the implicit Runge-Kutta scheme are employed to obtain a highly accurate approximation to the mentioned problem. J-GR-C method, based on Jacobi polynomials and Gauss-Radau quadrature integration, reduces solving the system of nonlinear hyperbolic equations to solve a system of nonlinear ordinary differential equations (SNODEs). In the examples given, numerical results by the J-GR-C method are compared with the exact solutions. In fact, by selecting relatively few J-GR-C points, we are able to get very accurate approximations. In this way, the results show that this method has a good accuracy and efficiency for solving coupled partial differential equations.

Aphmesh refinement method for optimal control

Summary A mesh refinement method is described for solving a continuous-time optimal control problem using collocation at Legendre–Gauss–Radau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative error estimate is derived based on the difference between the Lagrange polynomial approximation of the state and a Legendre–Gauss–Radau quadrature integration of the dynamics within a mesh interval. The derived relative error estimate is then used to decide if the degree of the approximating polynomial within a mesh should be increased or if the mesh interval should be divided into subintervals. The degree of the approximating polynomial within a mesh interval is increased if the polynomial degree estimated by the method remains below a maximum allowable degree. Otherwise, the mesh interval is divided into subintervals. The process of refining the mesh is repeated until a specified relative error tolerance is met. Three examples highlight various features of the method and show that the approach is more computationally efficient and produces significantly smaller mesh sizes for a given accuracy tolerance when compared with fixed-order methods. Copyright © 2014 John Wiley & Sons, Ltd.

A generalized framework for nodal first derivative summation-by-parts operators

A generalized framework is presented that extends the classical theory of finite-difference summation-by-parts (SBP) operators to include a wide range of operators, where the main extensions are (i) non-repeating interior point operators, (ii) nonuniform nodal distribution in the computational domain, (iii) operators that do not include one or both boundary nodes. Necessary and sufficient conditions are proven for the existence of nodal approximations to the first derivative with the SBP property. It is proven that the positive-definite norm matrix of each SBP operator must be associated with a quadrature rule; moreover, given a quadrature rule there exists a corresponding SBP operator, where for diagonal-norm SBP operators the weights of the quadrature rule must be positive. The generalized framework gives a straightforward means of posing many known approximations to the first derivative as SBP operators; several are surveyed, such as discontinuous Galerkin discretizations based on the Legendre–Gauss quadrature points, and shown to be SBP operators. Moreover, the new framework provides a method for constructing SBP operators by starting from quadrature rules; this is illustrated by constructing novel SBP operators from known quadrature rules. To demonstrate the utility of the generalization, the Legendre–Gauss and Legendre–Gauss–Radau quadrature points are used to construct SBP operators that do not include one or both boundary nodes.

Potential evaluation in space–time BIE formulations of 2D wave equation problems

In this paper we examine the computation of the potential generated by space–time BIE representations associated with Dirichlet and Neumann problems for the 2D wave equation. In particular, we consider the efficient evaluation of the (convolution) time integral that appears in the potential representation. For this, we propose two simple quadrature rules which appear more efficient than the currently used ones. Both are of Gaussian type: the classical Gauss–Jacobi quadrature rule in the case of a Dirichlet problem, and the classical Gauss–Radau quadrature rule in the case of a Neumann problem. Both of them give very accurate results by using a few quadrature nodes, as long as the potential is evaluated at a point not very close to the boundary of the PDE problem domain. To deal with this latter case, we propose an alternative rule, which is defined by a proper combination of a Gauss–Legendre formula with one of product type, the latter having only 5 nodes.The proposed quadrature formulas are compared with the second order BDF Lubich convolution quadrature formula and with two higher order Lubich formulas of Runge–Kutta type. An extensive numerical testing is presented. This shows that the new proposed approach is very competitive, from both the accuracy and efficiency points of view.

On the remainder term of Gauss–Radau quadrature with Chebyshev weight of the third kind for analytic functions

For analytic functions the remainder term of quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points ∓1 and a sum of semi-axes ρ>1, for Gauss–Radau quadrature formula with Chebyshev weight function of the third kind. Starting from the explicit expression of the corresponding kernel, derived by Gautschi, we determine the locations on the ellipses where maximum modulus of the kernel is attained. The obtained values confirm the corresponding conjectured values given by Gautschi in his paper [W. Gautschi, On the remainder term for analytic functions of Gauss–Lobatto and Gauss–Radau quadratures, Rocky Mounatin J. Math. 21 (1991) 209-206]. In this way the last unproved conjecture from the mentioned paper is now verified.

Quadratures associated with pseudo-orthogonal rational functions on the real half line with poles in [formula omitted

We consider a positive measure on [0,∞) and a sequence of nested spaces ℒ0⊂ℒ1⊂ℒ2⋯ of rational functions with prescribed poles in [−∞,0]. Let {φk}k=0∞, with φ0∈ℒ0 and φk∈ℒk∖ℒk−1, k=1,2,… be the associated sequence of orthogonal rational functions. The zeros of φn can be used as the nodes of a rational Gauss quadrature formula that is exact for all functions in ℒn⋅ℒn−1, a space of dimension 2n. Quasi- and pseudo-orthogonal functions are functions in ℒn that are orthogonal to some subspace of ℒn−1. Both of them are generated from φn and φn−1 and depend on a real parameter τ. Their zeros can be used as the nodes of a rational Gauss–Radau quadrature formula where one node is fixed in advance and the others are chosen to maximize the subspace of ℒn⋅ℒn−1 where the quadrature is exact. The parameter τ is used to fix a node at a preassigned point. The space where the quadratures are exact has dimension 2n−1 in both cases but it is in ℒn−1⋅ℒn−1 in the quasi-orthogonal case and it is in ℒn⋅ℒn−2 in the pseudo-orthogonal case.

The error norm of Gauss-Radau quadrature formulae for Chebyshev weight functions

In certain spaces of analytic functions the error term of the Gauss-Radau quadrature formula relative to a (nonnegative) weight function is a continuous linear functional. We compute or estimate the norm of the error functional for any one of the four Chebyshev weight functions.

Matrices, moments, and rational quadrature

Many problems in science and engineering require the evaluation of functionals of the form Fu(A)=uTf(A)u, where A is a large symmetric matrix, u a vector, and f a nonlinear function. A popular and fairly inexpensive approach to determining upper and lower bounds for such functionals is based on first carrying out a few steps of the Lanczos procedure applied to A with initial vector u, and then evaluating pairs of Gauss and Gauss–Radau quadrature rules associated with the tridiagonal matrix determined by the Lanczos procedure. The present paper extends this approach to allow the use of rational Gauss quadrature rules.

On the remainder term of Gauss–Radau quadratures for analytic functions

For analytic functions the remainder term of Gauss–Radau quadrature formulae can be represented as a contour integral with a complex kernel. We study the kernel on elliptic contours with foci at the points ± 1 and a sum of semi-axes ϱ > 1 for the Chebyshev weight function of the second kind. Starting from explicit expressions of the corresponding kernels the location of their maximum modulus on ellipses is determined. The corresponding Gautschi's conjecture from [On the remainder term for analytic functions of Gauss–Lobatto and Gauss–Radau quadratures, Rocky Mountain J. Math. 21 (1991), 209–226] is proved.

Gauss quadrature formula: An extension via interpolating orthogonal polynomials

The n -point Gauss quadrature rule states that ∫ - 1 1 f ( x ) ω ( x ) d x = ∑ i = 1 n w i f ( z i ) + R n ( f ) , where z i and w i , i = 1 , … , n , are called, respectively, the Gaussian nodes and weights. It is known that the formula is exact of degree 2 n - 1 . We provide an extension of this rule by considering x = - 1 and 1 as the pre-assigned nodes of certain order n 1 and n 2 , respectively. For this, we construct interpolating orthogonal polynomials that make the suggested rule capable of utilizing the maximum information related to the value and derivatives of the integrand f at these points. Our proposed rule is different from Gauss–Lobatto and Gauss–Radau quadrature formulae, which also take care of these points to a certain extent. The results related to the degree of exactness and convergence are also presented. Some questions related to our proposed rule which may require further investigation are narrated as well.

Plastic Hinge Integration Methods for Force-Based Beam–Column Elements

A new plastic hinge integration method overcomes the problems with nonobjective response caused by strain-softening behavior in force-based beam-column finite elements. The integration method uses the common concept of a plastic hinge length in a numerically consistent manner. The method, derived from the Gauss-Radau quadrature rule, integrates deformations over specified plastic hinge lengths at the ends of the beam-column element, and it has the desirable property that it reduces to the exact solution for linear problems. Numerical examples show the effect of plastic hinge integration on the response of force-based beam-column elements for both strain-hardening and strain-softening section behavior in the plastic hinge regions. The incorporation of a plastic hinge length in the element integration method ensures objective element and section response, which is important for strain-softening behavior in reinforced concrete structures. Plastic rotations are defined in a consistent manner and clearly related to deformations in the plastic hinges.

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.