In this paper a combination of discontinuous, piecewise linear, finite elements with implicitexplicit time stepping is considered for convection-reaction equations. Combined with low order quadrature rules, this leads to convenient schemes. We shall consider the effect of such low order quadrature rules on accuracy and stability for one-dimensional problems. 2000 Mathematics Subject Classificat...

Inequalities are obtained for quadrature rules in terms of upper and lower bounds of the first derivative of the integrand. Bounds of Ostrowski type quadrature rules are obtained and the classical Iyengar inequality for the trapezoidal rule is recaptured as a special case. Applications to numerical integration are demonstrated.

Quadrature rules on semi-infinite and infinite intervals are considered involving weight functions of the Laguerre and Hermite type. It is shown that such quadrature rules cannot have equal coefficients and real nodes unless the algebraic degree of accuracy is severely limited.

Two quadrature rules for the approximate evaluation of Cauchy principal value integrals, with nodes at the zeros of appropriate orthogonal polynomials, are discussed. An expression for the truncation error, in terms of higher order derivatives, is given for each rule. In addition, two theorems, containing sufficient conditions for the convergence of the sequence of quadrature rules to the integ...

Recently Laurie presented a fast algorithm for the computation of (2n + 1)-point Gauss-Kronrod quadrature rules with real nodes and positive weights. We describe modifications of this algorithm that allow the computation of Gauss-Kronrod quadrature rules with complex conjugate nodes and weights or with real nodes and positive and negative weights.

Given a positive bounded Borel measure μ on the interval [−1, 1], we provide convergence results in Lμ2 -norm to a function f of its sequence of rational interpolating functions at the nodes of rational Gauss-type quadrature formulas associated with the measure μ. As an application, we construct rational interpolatory quadrature formulas for complex bounded measures σ on the interval, and give ...

We study the optimal general rate of convergence of the n-point quadrature rules of Gauss and Clenshaw–Curtis when applied to functions of limited regularity: if the Chebyshev coefficients decay at a rate O(n−s−1) for some s > 0, Clenshaw–Curtis and Gauss quadrature inherit exactly this rate. The proof (for Gauss, if 0 < s < 2, there is numerical evidence only) is based on work of Curtis, Johns...

The purpose of this paper is to reduce the complexity computing components integral Fm-transform, m≥0, whose analytic expressions include definite integrals. We propose use nontrivial quadrature rules with nonuniformly distributed integration points instead widely used Newton–Cotes formulas. As weight function that determines orthogonality, we choose generating fuzzy partition associated Fm-tra...