Gauss legendre quadrature formulas over a tetrahedron

2005 ◽  
Vol 22 (1) ◽  
pp. 197-219 ◽  
Author(s):  
H. T. Rathod ◽  
B. Venkatesudu ◽  
K. V. Nagaraja
Keyword(s):  
Quadrature Formulas ◽  
Legendre Quadrature

scholarly journals Asymptotic Gauss Quadrature Errors as Fourier Coefficients of the Integrand

1971 ◽  
Vol 12 (3) ◽  
pp. 315-322 ◽  
Author(s):  
M. M. Chawla
Keyword(s):  
Complex Variable ◽  
Fourier Coefficients ◽  
Integral Representations ◽  
Quadrature Formulas ◽  
Orthogonal Expansion ◽  
Complex Variable Methods ◽  
Fourier Expansions ◽  
Gauss Type ◽  
Legendre Quadrature ◽  
Class Of Functions

The purpose of this paper is to derive asymptotic relations giving the error of a Gauss type quadrature, applied to analytic functions, in terms of certain coefficients in the orthogonal expansion of the integrand. The Fourier expansions of the integrand we consider here are those in terms of the Legendre and the Chebyshev polynomials. In Section 3 we obtain the error of the Gauss-Legendre quadrature expressed in terms of the Legendre-Fourier coefficients of the integrand. In Section 4 the errors of Gauss-Legendre, Lobatto and Radau quadrature formulas are obtained, for large n, expressed in terms of the Chebyshev-Fourier coefficients of the integrand. In deriving these estimates we have used complex variable methods restricting ourselves to the class of analytic integrands; this allows us to obtain simple contour integral representations for the errors of these quadratures for large values of n. However, the form of the estimates obtained indicate that these are applicable to a much wider class of functions.

scholarly journals Symplectic P-stable additive Runge—Kutta methods

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Antonella Zanna
Keyword(s):  
Variational Formulation ◽  
Quadrature Formulas ◽  
Symplectic Methods ◽  
Primary Method ◽  
Highly Oscillatory ◽  
Runge Kutta ◽  
Highly Oscillatory Problems ◽  
Symplectic Pair ◽  
Legendre Quadrature ◽  
Secondary Method

<p style='text-indent:20px;'>Classical symplectic partitioned Runge–Kutta methods can be obtained from a variational formulation where all the terms in the discrete Lagrangian are treated with the same quadrature formula. We construct a family of symplectic methods allowing the use of different quadrature formulas (primary and secondary) for different terms of the Lagrangian. In particular, we study a family of methods using Lobatto quadrature (with corresponding Lobatto IIIA-B symplectic pair) as a primary method and Gauss–Legendre quadrature as a secondary method. The methods have the same implicitness as the underlying Lobatto IIIA-B pair, and, in addition, they are <i>P-stable</i>, therefore suitable for application to highly oscillatory problems.</p>

On Quadrature Formulas

2018 ◽  
Vol 481 (2) ◽  
pp. 136-137
Author(s):  
V. Chubarikov ◽  
Keyword(s):  
Quadrature Formulas

scholarly journals Collocation Methods for High-Order Well-Balanced Methods for Systems of Balance Laws

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1799
Author(s):  
Irene Gómez-Bueno ◽  
Manuel Jesús Castro Díaz ◽  
Carlos Parés ◽  
Giovanni Russo
Keyword(s):  
High Order ◽  
Collocation Methods ◽  
Quadrature Formulas ◽  
Balance Laws ◽  
Source Terms ◽  
Time Step ◽  
Systems Of Balance Laws ◽  
Reconstruction Operator ◽  
Non Linear ◽  
Linear Problems

In some previous works, two of the authors introduced a technique to design high-order numerical methods for one-dimensional balance laws that preserve all their stationary solutions. The basis of these methods is a well-balanced reconstruction operator. Moreover, they introduced a procedure to modify any standard reconstruction operator, like MUSCL, ENO, CWENO, etc., in order to be well-balanced. This strategy involves a non-linear problem at every cell at every time step that consists in finding the stationary solution whose average is the given cell value. In a recent paper, a fully well-balanced method is presented where the non-linear problems to be solved in the reconstruction procedure are interpreted as control problems. The goal of this paper is to introduce a new technique to solve these local non-linear problems based on the application of the collocation RK methods. Special care is put to analyze the effects of computing the averages and the source terms using quadrature formulas. A general technique which allows us to deal with resonant problems is also introduced. To check the efficiency of the methods and their well-balance property, they have been applied to a number of tests, ranging from easy academic systems of balance laws consisting of Burgers equation with some non-linear source terms to the shallow water equations—without and with Manning friction—or Euler equations of gas dynamics with gravity effects.

Sign in / Sign up

Export Citation Format

Share Document