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...

We propose a two algorithms for computation of (sharp) enclosures of definite interevals: a lor.rd adaptive dgorid~a (LAA) and a g/oba/ a&,pth~ cdgor92~n (GAA). Both algorithms are based (m Gauss-Legendre ~ladrature. Error terms are bounded using automatic differentiation in combination with interval evah:adons. Several nmnerical examples are presented; these examples include comparis~m with an...

In this paper, a simple scheme is constructed for finding approximate solution of the nonlinear Fredholm integral equation second kind. To end, Lagrange interpolation polynomials together with Gauss-Legendre quadrature rule are used to transform source problem system algebraic equations. Afterwards, resulting can be solved by Newton method. The basic idea choose points same as integration. This...

This paper develops a shock capturing approach for high-order correction procedure via reconstruction (CPR) method with Legendre-Gauss solution points. Shock regions are treated by novel compact nonuniform nonlinear weighted (CNNW) schemes, which have the same points as CPR method. CNNW schemes constructed discretizing flux derivatives based on Riemann fluxes at in one cell and using (NNW) inte...

Let p be an odd prime and {χ(m) = (m/p)}, m = 0,1, . . . ,p − 1 be a finite arithmetic sequence with elements the values of a Dirichlet character χ modp which are defined in terms of the Legendre symbol (m/p), (m,p)= 1. We study the relation between the Gauss and the quadratic Gauss sums. It is shown that the quadratic Gauss sumsG(k;p) are equal to the Gauss sums G(k,χ) that correspond to this ...

Nodal point sets, and associated collocation projections, play an important role in a range of high-order methods, including Flux Reconstruction (FR) schemes. Historically, efforts have focused on identifying nodal sets that aim to minimise the L∞ error interpolating polynomial. The present work combines comprehensive review known approximation theory results, with new numerical experiments, mo...

Abstract We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes iff n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p−1 which is the product of a power of p and the generating polynomial of the partial sums of Legendr...

In this paper, we consider numerical differentiation of bivariate functions when a set of noisy data is given. A mollification method based on spanned by Legendre polynomials is proposed and the mollification parameter is chosen by a discrepancy principle. The theoretical analyses show that the smoother the genuine solution, the higher the convergence rates of the numerical solution. To get a p...

Other machines on which the adaption has been tested: CRAY-i Other machines on which the adaption has been tested: CRAY-i and 1BM4341/2 No. of bits in a byte: 8 No. of bits in a byte: 8 No. of lines required to effect adaptation: 190 No. of lines required to effect adaptation: 215 Additional keywords: spherical oscillator functions . Additional keywords: Gauss—Legendre quadrature, isotopic subs...