Numerical integration is a common sub-problem in many applications. It can be solved easily in CPU-based applications using adaptive quadrature such as the adaptive Simpson’s rule. These algorithms rely, however, on error estimation yielding a significant computational overhead. In addition, they require recursive function evaluations, which are not well suited for parallel computation on graph...

Numerical approximations of definite integrals and related error estimations can be made using Simpson’s rules (inequalities). There are two well-known rules: 13 rule or quadrature formula 38 second formula. The aim the present paper is to extend several inequalities that hold for rule. More precisely, we prove a weighted version type inequality some Lipschitzian, bounded variations, convex fun...

We introduce an accurate cell-centered method for modeling Darcy flow on general quadrilateral, hexahedral, and simplicial grids. We refer to these discretizations as the multipoint-flux mixed-finiteelement (MFMFE) method. The MFMFE method is locally conservative with continuous fluxes and can be viewed within a variational framework as a mixed finite-element method with special approximating s...

Let // be the integral of f(x) over an A^-dimensional hypercube and Q(m>f be the approximation to //obtained by subdividing the hypercube into m equal subhypercubes and applying the same quadrature rule Q to each. In order to extrapolate efficiently for // on the basis of several different approximations Q ' /, it is necessary to know the form of the error functional Q\m'f If as an expansion in...

We develop a local flux mimetic finite difference method for second order elliptic equations with full tensor coefficients on polyhedral meshes. To approximate the velocity (vector variable), the method uses two degrees of freedom per element edge in two dimensions and n degrees of freedom per n-gonal mesh face in three dimensions. To approximate the pressure (scalar variable), the method uses ...

-In this paper, the asymptotic bit operation cost of a family of quadrature formulas, especially suitable for evaluation of improper integrals, is studied. More precisely, we consider the family of quadrature formulas obtained by applying k times the variable transformation x : sinh(y) and then the trapezoidal rule to the transformed integral. We prove that, if the integrand function is analyti...

Direct numerical solution of the coordinate-space integral-equation version of the two-particle Lippmann-Schwinger (LS) equation is considered without invoking the traditional partial-wave decomposition. The singular kernel of the three-dimensional LS equation in coordinate space is regularized by a subtraction technique. The resulting nonsingular integral equation is then solved via the Nystro...

In [5] and [2] the authors describe the remarkable effectiveness of the doubly exponential ‘tanh-sinh’ transformation for numerical integration —even for quite unruly integrands. Our intention in this note is to provide a theoretical underpinning when the integrand is analytic for the observed superlinear convergence of the method. Our analysis rests on the corresponding but somewhat easier ana...

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...