Accurate calculation of the solutions to the Thomas-Fermi equations

We obtain highly accurate solutions to the Thomas-Fermi equations for atoms and atoms in very strong magnetic fields. We apply the Pad\'e-Hankel method, numerical integration, power series with Pad\'e and Hermite-Pad\'e 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

Chebyshev expansion on intervals with branch points with application to the root of Kepler’s equation: A Chebyshev–Hermite–Padé method

AbstractWhen two or more branches of a function merge, the Chebyshev series of u(λ) will converge very poorly with coefficients an of Tn(λ) falling as O(1/nα) for some small positive exponent α. 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 u1(λ) and u2(λ), then both branches can be written without approximation as the roots of (u−u1(λ))(u−u2(λ))=u2+β(λ)u+γ(λ). By expanding the nonsingular coefficients of the quadratic, β(λ) and γ(λ), 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(−σN) for some constant σ>0 where N is the truncation of the Chebyshev series. This is dubbed the “Chebyshev–Shafer” or “Chebyshev–Hermite–Padé” method because it substitutes Chebyshev series for power series in the generalized Padé approximants known variously as “Shafer” or “Hermite–Padé” 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;λ,ϵ)≡u−ϵsin(u)−λ=0. Only one of the merging roots is real-valued and the root depends on two parameters (λ,ϵ) rather than one. Nonetheless, the Chebyshev–Hermite–Padé 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

Error saturation in Gaussian radial basis functions on a finite interval

AbstractRadial basis function (RBF) interpolation is a “meshless” strategy with great promise for adaptive approximation. One restriction is “error saturation” which occurs for many types of RBFs including Gaussian RBFs of the form ϕ(x;α,h)=exp(−α2(x/h)2): in the limit h→0 for fixed α, the error does not converge to zero, but rather to ES(α). Previous studies have theoretically determined the saturation error for Gaussian RBF on an infinite, uniform interval and for the same with a single point omitted. (The gap enormously increases ES(α).) We show experimentally that the saturation error on the unit interval, x∈[−1,1], is about 0.06exp(−0.47/α2)‖f‖∞ — huge compared to the O(2π/α2)exp(−π2/[4α2]) saturation error for a grid with one point omitted. We show that the reason the saturation is so large on a finite interval is that it is equivalent to an infinite grid which is uniform except for a gap of many points. The saturation error can be avoided by choosing α≪1, the “flat limit”, but the condition number of the interpolation matrix explodes as O(exp(π2/[4α2])). The best strategy is to choose the largest α which yields an acceptably small saturation error: If the user chooses an error tolerance δ, then αoptimum(δ)=1/−2log(δ/0.06)

Multiprocessing in Meteorological Models

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/94918/1/eost7470.pd

Bound states in weakly deformed waveguides: numerical vs analytical results

We have studied the emergence of bound states in weakly deformed and/or heterogeneous waveguides, comparing the analytical predictions obtained using a recently developed perturbative method, with precise numerical results, for different configurations (a homogeneous asymmetric waveguide, a heterogenous asymmetric waveguide and a homogeneous broken-strip). In all the examples considered in this paper we have found excellent agreement between analytical and numerical results, thus providing a numerical verification of the analytical approach.Comment: 11 pages, 6 figure

Asymptotic Coefficients and Errors for Chebyshev Polynomial Approximations with Weak Endpoint Singularities: Effects of Different Bases

When solving differential equations by a spectral method, it is often convenient to shift from Chebyshev polynomials $T_{n}(x)$ with coefficients $a_{n}$ to modified basis functions that incorporate the boundary conditions. For homogeneous Dirichlet boundary conditions, $u(\pm 1)=0$, popular choices include the ``Chebyshev difference basis", $\varsigma_{n}(x) \equiv T_{n+2}(x) - T_{n}(x)$ with coefficients here denoted $b_{n}$ and the ``quadratic-factor basis functions" $\varrho_{n}(x) \equiv (1-x^{2}) T_{n}(x)$ with coefficients $c_{n}$. If $u(x)$ is weakly singular at the boundaries, then $a_{n}$ will decrease proportionally to $\mathcal{O}(A(n)/n^{\kappa})$ for some positive constant $\kappa$, where the $A(n)$ is a logarithm or a constant. We prove that the Chebyshev difference coefficients $b_{n}$ decrease more slowly by a factor of $1/n$ while the quadratic-factor coefficients $c_{n}$ decrease more slowly still as $\mathcal{O}(A(n)/n^{\kappa-2})$. The error for the unconstrained Chebyshev series, truncated at degree $n=N$, is $\mathcal{O}(|A(N)|/N^{\kappa})$ in the interior, but is worse by one power of $N$ in narrow boundary layers near each of the endpoints. Despite having nearly identical error \emph{norms}, the error in the Chebyshev basis is concentrated in boundary layers near both endpoints, whereas the error in the quadratic-factor and difference basis sets is nearly uniform oscillations over the entire interval in $x$. Meanwhile, for Chebyshev polynomials and the quadratic-factor basis, the value of the derivatives at the endpoints is $\mathcal{O}(N^{2})$, but only $\mathcal{O}(N)$ for the difference basis