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We study the rotations of a heavy string (helicoseir) about a vertical axis with one free endpoint and with arbitrary density, under the action of the gravitational force. We show that the problem can be transformed into a nonlinear eigenvalue equation, as in the uniform case. The eigenmodes of this equation represent equilibrium configurations of the rotating string in which the shape of the string doesn't change with time. As previously proved by Kolodner for the homogenous case, the occurrence of new modes of the nonlinear equation is tied to the spectrum of the corresponding linear equation. We have been able to generalize this result to a class of densities $\rho(s) = \gamma (1-s)^{\gamma-1}$, which includes the homogenous string as a special case ($\gamma=1$). We also show that the solutions to the nonlinear eigenvalue equation (NLE) for an arbitrary density are orthogonal and that a solution of this equation with a given number of nodes contains solutions of a different helicoseir, with a smaller number of nodes. Both properties hold also for the homogeneous case and had not been established before., Comment: 13 pages, 3 figures

Hough functions are the eigenfunctions of the Laplace tidal equation governing fluid motion on a rotating sphere with a resting basic state. Several numerical methods have been used in the past. In this paper, we compare two of those methods: normalized associated Legendre polynomial expansion and Chebyshev collocation. Both methods are not widely used, but both have some advantages over the commonly used unnormalized associated Legendre polynomial expansion method. Comparable results are obtained using both methods. For the first method we note some details on numerical implementation. The Chebyshev collocation method was first used for the Laplace tidal problem by Boyd (1976) and is relatively easy to use. A compact MATLAB code is provided for this method. We also illustrate the importance and effect of including a parity factor in Chebyshev polynomial expansions for modes with odd zonal wave numbers.

A “bent waveguide” in the sense used here is a small perturbation of a two-dimensional rectangular strip which is infinitely long in the down-channel direction and has a finite, constant width in the cross-channel coordinate. The goal is to calculate the smallest (“ground state”) eigenvalue of the stationary Schrödinger equation which here is a two-dimensional Helmholtz equation, ψxx+ψyy+Eψ=0 where E is the eigenvalue and homogeneous Dirichlet boundary conditions are imposed on the walls of the waveguide. Perturbation theory gives a good description when the “bending strength” parameter ϵ is small as described in our previous article (Amore et al., 2017) and other works cited therein. However, such series are asymptotic, and it is often impractical to calculate more than a handful of terms. It is therefore useful to develop numerical methods for the perturbed strip to cover intermediate ϵ where the perturbation series may be inaccurate and also to check the pertubation expansion when ϵ is small. The perturbation-induced change-in-eigenvalue, δ≡E(ϵ)−E(0), is O(ϵ2). We show that the computation becomes very challenging as ϵ→0 because (i) the ground state eigenfunction varies on both O(1) and O(1∕ϵ) length scales and (ii) high accuracy is needed to compute several correct digits in δ, which is itself small compared to the eigenvalue E. The multiple length scales are not geographically separate, but rather are inextricably commingled in the neighborhood of the boundary deformation. We show that coordinate mapping and immersed boundary strategies both reduce the computational domain to the uniform strip, allowing application of pseudospectral methods on tensor product grids with tensor product basis functions. We compared different basis sets; Chebyshev polynomials are best in the cross-channel direction. However, sine functions generate rather accurate analytical approximations with just a single basis function. In the down-channel coordinate, X∈[−∞,∞], Fourier domain truncation using the change of coordinate X=sinh(Lt) is considerably more efficient than rational Chebyshev functions TBn(X;L). All the spectral methods, however, yielded the required accuracy on a desktop computer. Fil: Boyd, John P.. University of Michigan; Estados Unidos Fil: Amore, Paolo. Universidad de Colima; México Fil: Fernández, Francisco Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas; Argentina

The classical boundary layer problem formulated by Heinrich Blasius more than a century ago is revisited, with the purpose of deriving simple and accurate analytical approximations to its solution. This is achieved through the combined use of a generalized Padé approach and of an integral iteration scheme devised by Hermann Weyl. The iteration scheme is also used to derive very accurate bounds for the value of the second derivative of the Blasius function at the origin, which plays a crucial role in this problem. © 2015 Elsevier B.V. All rights reserved.

We obtain highly accurate solutions to the Thomas-Fermi equations for atoms and atoms in very strong magnetic fields. We apply the Padé-Hankel method, numerical integration, power series with Padé and Hermite-Padé approximants and Chebyshev polynomials. Both the slope at origin and the location of the right boundary in the magnetic-field case are given with unprecedented accuracy., Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich. This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively. A bis-μ-amidoximato-bridged cobalt(III) dimer obtained with a new tetrapodal ligand possesses interesting structural parameters as a consequence of intramolecular hydrogen bonding intentionally built into the complex. Its synthesis and properties are described. The new ligand type combines attributes of two previously described ligand classes: It binds a metal ion in a tetrapodal pentadentate fashion and forms a pseudomacrocycle through hydrogen bonds, characteristic of chelating oxime ligands. Coordination isomerism, which is a consequence of dimer formation, has been analyzed by means of X-ray crystallography and carbon-13 nuclear magnetic resonance spectroscopy.

We show how local approximations, each accurate on a subinterval, can be blended together to form a global approximation which is accurate over the entire interval. The blending functions are smoothed approximations to a step function, constructed using the error function. The local approximations may be power series, asymptotic expansion, or other more exotic species. As an example, for the dilogarithm function, we construct a one-line analytic approximation which is accurate to one part in 700. This can be generalized to higher order merely by adding more terms in the local approximations. We also show the failure of the alternative strategy of subtracting singularities.

Seven types of Chebyshev-like grids in one dimension are compared according to four different criteria for accuracy. The grid which minimizes the Lebesgue constant is the best because it performs fairly well by all four criteria. For the same reason, minimizing the Lebesgue constant seems to be the most useful measure of optimality because grids that are good according to this criterion are good when measured by other criteria, too. CFM-optimality, which is the property that all cardinal functions (Lagrange fundamental polynomials) have maxima at the interpolation points, seems to be the least-discriminating criterion because all seven grids generate cardinal functions that have maxima at or very near the interpolation points. The difference between these grids on all four criteria, always less than a factor of two and usually much smaller, are sufficiently modest so that the final choice between grids should probably not be made because of accuracy, but rather based on other criteria such as ease-of-programming, analytical simplicity, conformality with other approximations, and timestep in applications to partial differential equations. To explore such nonaccuracy issues, we also compared six of the grids on the basis of maximum allowed timestep when the grid is used to discretize the spatial coordinate and an explicit scheme is used for time-marching. The Lebesgue-optimal grid is also nearly optimal in the sense of allowing the longest timestep. With a pseudospectral (collocation) algorithm, the Lebesgue grid allows a timestep three-halves as long as that of the Legendre-Lobatto spectral element method.

Radial basis function (RBF) interpolation is a ''meshless'' strategy with great promise for adaptive approximation. Because it is meshless, there is no canonical grid to act as a starting point for function-adaptive grid modification. Uniform grids are therefore common with RBFs. Like polynomial interpolation, which is the limit of RBF interpolation when the RBF functions are very wide (the ''flat limit''), RBF interpolants are vulnerable to the Runge Phenomenon, whether the grid is uniform or irregular: the N-point interpolant may diverge as N->~ on the interval spanned by the interpolation points even for a function f(x) that is analytic everywhere on and near this interval. In this work, we discuss six strategies for defeating the Runge Phenomenon, specializing to a uniform grid on a finite interval and one particular species of RBFs in one space dimension: @f(x)=exp(-[@a^2/h^2]x^2) where h is the grid spacing. Three strategies fail, but three are successful. One good strategy is to vary @a with the number of interpolation points N as N^-^1^/^4. Unfortunately this gives both a subgeometric rate of convergence for the approximation (error falling as exp(-qN) for some constant q) and a matrix condition number that grows as exp(pN) for some p>0. (In order to explain why fixed@a, independent of N, is not a convergent strategy, we digress to discuss error saturation, and show experimentally that the saturation error on the finite interval is about 0.6exp(-0.47/@a^2)@?f@?"~; if a user-chosen error tolerance @d is acceptable, then the optimum choice is @a"o"p"t"i"m"u"m(@d)=1/-2log(@d/0.06).) The second good strategy is RBF Extension, which uses RBF functions with centers on a interval slightly larger than the target interval to approximate f(x). A third strategy is to split the interval into two ''boundary layers'' and a large middle interval and apply separate RBF approximations on each. The slow-decrease-of-@a strategy is much cheaper, but RBF Extension is much more resistant to ill-conditioning and therefore can achieve much lower errors. The three-interval method is the least accurate, but is robust and inexpensive.

It is shown that the rational functions of Higgins and Christov, orthogonal on [−∞, ∞], are Chebyshev polynomials of the first and second kinds with an algebraic change of variable. Because of these relationships, the existing theory and algorithms for mapped Chebyshev polynomials also apply to the rational functions: the Higgins and Christov functions have excellent numerical properties. However —precisely because of these same connections—it is usually simpler to use the change of variable rather than write computer programs that employ the Higgins and Christov functions themselves. Nonetheless, the result is a series of orthogonal rational functions. For some problems whose solutions decay slowly (algebraically rather than exponentially with ¦y¦), such as the “Yoshida jet” in oceanography, a Christov expansion is the only spectral series that converges rapidly.

Implicit timestepping schemes and the Nonlinear Galerkin (NG) algorithms are com- petitors for integrating on the slow manifold, that is, for computing solutions when the timestep- limiting high frequencies are physically unimportant and most of the energy is in low frequency flow. We show that implicit schemes are always at least as accurate as the lowest nontrivial NG method, and can be made superior to even high order NG schemes by shortening the time step. where ak(t) is the k th Fourier or normal mode coefficient, Wk(t) is the frequency of the kth mode, and d(t) is the lath component of the sum of the nonlinear terms and the external forcing (I), and subscript t denotes time differentiation. Examples, with the spectral basis {exp(ilcz)}, include the Korteweg-deVries (KdV) equation, for which w = -k3, and Burgers' equation, for which

When two or more branches of a function merge, the Chebyshev series of u(@l) will converge very poorly with coefficients a"n of T"n(@l) falling as O(1/n^@a) for some small positive exponent @a. However, as shown in [J.P. Boyd, Chebyshev polynomial expansions for simultaneous approximation of two branches of a function with application to the one-dimensional Bratu equation, Appl. Math. Comput. 143 (2002) 189-200], it is possible to obtain approximations that converge exponentially fast in n. If the roots that merge are denoted as u"1(@l) and u"2(@l), then both branches can be written without approximation as the roots of (u-u"1(@l))(u-u"2(@l))=u^[email protected](@l)[email protected](@l). By expanding the nonsingular coefficients of the quadratic, @b(@l) and @c(@l), as Chebyshev series and then applying the usual roots-of-a-quadratic formula, we can approximate both branches simultaneously with error that decreases proportional to exp([email protected]) for some constant @s>0 where N is the truncation of the Chebyshev series. This is dubbed the ''Chebyshev-Shafer'' or ''Chebyshev-Hermite-Pade'' method because it substitutes Chebyshev series for power series in the generalized Pade approximants known variously as ''Shafer'' or ''Hermite-Pade'' approximants. Here we extend these ideas. First, we explore square roots with branches that are both real-valued and complex-valued in the domain of interest, illustrated by meteorological baroclinic instability. Second, we illustrate triply branched functions via roots of the Kepler equation, f(u;@l,@e)[email protected](u)[email protected]=0. Only one of the merging roots is real-valued and the root depends on two parameters (@l,@e) rather than one. Nonetheless, the Chebyshev-Hermite-Pade scheme is successful over the whole two-dimensional parameter plane. We also discuss how to cope with poles and logarithmic singularities that arise in our examples at the extremes of the expansion domain.

Runge showed that polynomial interpolation using an equispaced grid often diverges as the degree N of the interpolating polynomial f X increases, even when f(x) is analytic over the whole interval. We suppress Runge divergence by defining the approximating polynomial as the minimizer of the sum of the interpolation residual plus a constant - times a smoothness norm. The Tikhonov parameter - can be determined easily through the method of the L-shaped curve. The resulting “Tikhonov approximant” is at least as accurate as the truncated, N th degree Chebyshev series for the same function.