It is well known that semi-discrete high order discontinuous Galerkin (DG) methods satisfy cell entropy inequalities for the square entropy for both scalar conservation laws and symmetric hyperbolic systems, in any space dimension and for any triangulations [39, 36]. However, this property holds only for the square entropy and the integrations in the DG methods must be exact. It is significantl...

Numerical quadrature is another name for numerical integration, which refers to the approximation of an integral ́ f (x)dx of some function f (x) by a discrete summation ∑wi f (xi) over points xi with some weights wi. There are many methods of numerical quadrature corresponding to different choices of points xi and weights wi, from Euler integration to sophisticated methods such as Gaussian quad...

Article history: Received 21 January 2010 Accepted 24 March 2010 Available online 17 April 2010 Submitted by R.A. Brualdi AMS classification: Primary: 65F50, 65F60, 15A16 Secondary: 05C20, 05C82

Denote by Pn =1 a f (x ) the Gaussian quadrature rule for the integral R 1 1 f (x) dx. We give a simple explicit expression for the \variance" Pn =1 a2 . The method can be used to obtain similar results for the Lobatto rule. 1 1. The result A quadrature rule (on [ 1; 1]) is a functional Qn on C [ 1; 1] of the form Qn [f ] = n X =1 a f (x ) ; a 2 R (1) 1 x1 < x2 < : : : < xn 1 : The \variance of...

Abstract We present a family of high-order trapezoidal rule-based quadratures for class singular integrals, where the integrand has point singularity. The part is expanded in Taylor series involving terms increasing smoothness. are based on rule, with quadrature weights Cartesian nodes close to singularity judiciously corrected expansion. High-order accuracy can be achieved by utilizing suffici...

A quadrature-rule type approximation of the quasi-continuum method for atomistic mechanics is presented. Simple analysis and computational experiments are presented that illustrate that the new method has, for the same accuracy, lower complexity compared to not only the quasicontinuum method, but also to cluster-type approximations of that method. A discussion about some implementation issues c...

A fractional trapezoidal rule type difference scheme for fractional order integro-differential equation is considered. The equation is discretized in time by means of a method based on the trapezoidal rule: while the time derivative is approximated by the standard trapezoidal rule, the integral term is discretized by means of a fractional quadrature rule constructed again from the trapezoidal r...

Inequalities are obtained for weighted integrals in terms of bounds involving the first derivative of the function. Quadrature rules are obtained and the classical Iyengar inequality for the trapezoidal rule is recaptured as a special case when the weight function w (x) ≡ 1. Applications to numerical integration are demonstrated.

In this paper, we suggest and analyze a new two-step iterative method for solving nonlinear fuzzy equations using the Trapezoidal quadrature rule. We prove that this method has quadratic convergence. The fuzzy quantities are presented in parametric form. Sever examples are given to illustrate the efficiency of the proposed method. Mathematics Subject Classification: 03E72; 37C25