We compare the convergence behavior of Gauss quadrature with that of its younger brother, Clenshaw–Curtis. Seven-line MATLAB codes are presented that implement both methods, and experiments show that the supposed factor-of-2 advantage of Gauss quadrature is rarely realized. Theorems are given to explain this effect. First, following O’Hara and Smith in the 1960s, the phenomenon is explained as ...

An efficient implementation of the high-order spectral volume (SV) method is presented for multidimensional conservation laws on unstructured grids. In the SV method, each simplex grid cell is called a spectral volume (SV), and the SV is further subdivided into polygonal (2D), or polyhedral (3D) control volumes (CVs) to support high-order data reconstructions. In the traditional implementation,...

In this chapter, we discuss some recently obtained asymptotic expansions related to problems in numerical analysis and approximation theory. • We present a generalization of the Euler–Maclaurin (E–M) expansion for the trapezoidal rule approximation of finite-range integrals R b a f ðxÞdx, when f(x) is allowed to have arbitrary algebraic–logarithmic endpoint singularities. We also discuss effect...

In the first part of this work, we derive compact numerical quadrature formulas for finite-range integrals I [f ] = ∫ b a f (x) dx, where f (x)= g(x)|x − t |β , β being real. Depending on the value of β, these integrals are defined either in the regular sense or in the sense of Hadamard finite part. Assuming that g ∈ C∞[a, b], or g ∈ C∞(a, b) but can have arbitrary algebraic singularities at x ...

FIGURE 2. Divergence of the arbitrary grid Legendre Galerkin method for improperly imposed initial conditions. Time-stable boundary conditions for nite-diierence schemes solving hyperbolic systems: methodology and application to high-order compact schemes, JCP 111, 220, (1994). 25 To initialize the problem, we must construct an approximation to the initial condition f(x), based on the grid poin...

A bstract We present analytic results of the two-loop master integrals for hadronic tW production that contain two massive propagators. For planar integral family, we succeed in constructing canonical basis, so are written terms multiple polylogarithms. The non-planar families have been calculated by Taylor series expansion around $$ {m}_W^2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/Math...

We derive an indefinite quadrature formula, based on a theorem of Ganelius, for Hp functions, for p > 1, over the interval (−1, 1). The main factor in the error of our indefinite quadrature formula is O(e−π √ ), with 2N nodes and 1 p + 1 q = 1. The convergence rate of our formula is better than that of the Stenger-type formulas by a factor of √ 2 in the constant of the exponential. We conjectur...

We express the cohomology of the basic Newton stratum of some unitary Shimura varieties associated to division algebras in terms of automorphic representations of the group in the Shimura datum. 2010 Mathematics Subject Classification: Primary 11G18 ; Secondary 11F70, 11F72.

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...

Let L be a quadratic imaginary field, inert at the rational prime p. Fix an integer n ≥ 3, and let M be the moduli space (in characteristic p) of principally polarized abelian varieties of dimension n equipped with an action by OL of signature (1, n−1). We show that each Newton stratum of M, other than the supersingular stratum, is irreducible.