Numerical Relativity in Spherical Polar Coordinates

A finite-difference method for solving three-dimensional, time-dependent incompressible Navier-Stokes equations in spherical polar coordinates is presented. A new algorithm, which is second-order accurate in time and space, is considered, and decoupling between the velocity and the pressure is achieved by this algorithm. Further, the numerical method is used to simulate the spherical Couette flow between two concentric spheres with the inner one rotating. A comparison of the numerical results with the available experimental measurements was made. It is demonstrated that the numerical code is valid for solving three-dimensional, unsteady incompressible Navier-Stokes equations in spherical polar coordinates.

Visibility precomputation has become essential for PRT-based rendering involving soft shadows. While graphic hardware rasterization has been increasingly used to compute visibility information for a mass of precomputation tasks, the linear feature of rasterization makes it unadaptive to sample visibility in a coordinates system, to which the Cartesian coordinates are non-linearly mapped. With the emergence of Geometry Shader in the framework of DirectX 10, this problem can be solved. In this paper, we present an efficient solution for visibility sampling on the spherical polar coordinates in the precomputation stage of PRT by means of the Geometry Shader within the DirectX 10 pipelines. In the mean time, a spherical reparameterization scheme is introduced for shadow rendering, under which spherical functions as lighting, BRDF and visibility are mapped onto the spherical polar coordinates.

We present a new three-dimensional general relativistic hydrodynamics code which is intended for simulations of stellar core collapse to a neutron star, as well as pulsations and instabilities of rotating relativistic stars. Contrary to the common approach followed in most existing three-dimensional numerical relativity codes which are based in Cartesian coordinates, in this code both the metric and the hydrodynamics equations are formulated and solved numerically using spherical polar coordinates. A distinctive feature of this new code is the combination of two types of accurate numerical schemes specifically designed to solve each system of equations. More precisely, the code uses spectral methods for solving the gravitational field equations, which are formulated under the assumption of the conformal flatness condition (CFC) for the three-metric. Correspondingly, the hydrodynamics equations are solved by a class of finite difference methods called high-resolution shock-capturing schemes, based upon state-of-the-art Riemann solvers and third-order cell-reconstruction procedures. We demonstrate that the combination of a finite difference grid and a spectral grid, on which the hydrodynamics and metric equations are, respectively, solved, can be successfully accomplished. This approach, which we call Mariage des Maillages (French for grid wedding), results in high accuracy of the metric solver and, in practice, allows for fully three-dimensional applications using computationally affordable resources, along with ensuring long-term numerical stability of the evolution. We compare our new approach to two other, finite difference based, methods to solve the metric equations which we already employed in earlier axisymmetric simulations of core collapse. A variety of tests in two and three dimensions is presented, involving highly perturbed neutron star spacetimes and (axisymmetric) stellar core collapse, which demonstrate the ability of the code to handle spacetimes with and without symmetries in strong gravity. These tests are also employed to assess the gravitational waveform extraction capabilities of the code, which is based on the Newtonian quadrupole formula. The code presented here is not limited to approximations of the Einstein equations such as CFC, but it is also well suited, in principle, to recent constrained formulations of the metric equations where elliptic equations have a preeminence over hyperbolic equations.

This paper considers the stabilization problem of a continuous linear system over a communication channel with network-induced delay. The channel is of finite data rate and connects the measurement sensor to the controller. Based on spherical polar coordinates, a novel coding scheme is proposed for this problem. In this coding scheme, the quantizer does not quantize the state of the original quantized system directly, but quantizes the state of the corresponding augmented system to produce control inputs, which simplifies the design of the coding scheme; a definite relation between the quantized data and the corresponding quantization error is used to facilitate the analysis of the stability of the system. Based on this coding scheme, the paper presents the condition guaranteeing the asymptotic stability of the system with network-induced delay and gives the design procedure of quantizer and controller. It is shown that adopting spherical polar coordinates makes the practical solving for the parameters of the quantizer and the controller numerically realistic.

This article shows that in spherical polar coordinates, some noncentral separable potentials have supersymmetry and shape invariance in the r and θ dimensions, we choose Hartmann potential and ring-shaped oscillator as two important examples, thus in principle the energy eigenvalues and energy eigenfunctions of such the potentials in the r and θ dimensions can be obtained by the method of supersymmetric quantum mechanics. Here we use an alternative method to get the required results.

Stabilization of a discrete time linear system over finite data-rate channel with noise attenuation performance by spherical polar coordinate quantizer

Optical coherence tomography (OCT) has become an important tool for measuring the vibratory response of the living cochlea. It stands alone in its capacity to measure the intricate motion of the hearing organ through the surrounding otic capsule bone. Nevertheless, as an extension of phase-sensitive OCT, it is only capable of measuring motion along the optical axis. Hence, measurements are 1-D. To overcome this limitation and provide a measure of the 3-D vector of motion in the cochlea, we developed an OCT system with three sample arms in a single interferometer. Taking advantage of the long coherence length of our swept laser, we depth (frequency) encode the three channels. An algorithm to depth decode and coregister the three channels is followed by a coordinate transformation that takes the vibrational data from the experimental coordinate system to Cartesian or spherical polar coordinates. The system was validated using a piezo as a known vibrating element that could be positioned at various angles. The angular measurement on the piezo was shown to have an RMSE of ≤ 0.30° (5.2 mrad) with a standard deviation of the amplitude of ≤ 120 pm. Finally, we demonstrate the system for in vivo imaging by measuring the vector of motion over a volume image in the apex of the mouse cochlea.

This paper considers an integral transform adapted to the solution of certain boundary value problems connected with the Helmholtz equation in cylindrical or spherical polar coordinates when the radial variable r varies over some infinite interval $0 < a \leqq r < \infty $. At infinity a radiation type limiting condition is imposed. A formula of inversion is derived which does not involve any summability factor, despite the singular nonself-adjoint nature of the expansion problem.

An exterior complex scaling technique is applied to compute Stark resonance parameters for two molecular orbitals ($1b_{1}$ and $1b_{2}$) represented in the field-free limit in a single-center expansion. For electric DC field configurations that guarantee azimuthal symmetry of the solution the calculation is carried out by solving a two-dimensional partial differential equation in spherical polar coordinates using a finite-element method. The resonance positions and widths as a function of electric field strengths are shown for field strengths starting in the tunnelling ionization regime, and extending well into the over-barrier ionization region.

We investigate the classical and quantum mechanics of diatomic molecules in noncollinear (tilted) static electric and nonresonant linearly polarized laser fields. The classical diatomic in tilted fields is a nonintegrable system, and we study the phase space structure for physically relevant parameter regimes for the molecule KCl. While exhibiting low-energy (pendular) and high-energy (free-rotor) integrable limits, the rotor in tilted fields shows chaotic dynamics at intermediate energies, and the degree of classical chaos can be tuned by changing the tilt angle. We examine the quantum mechanics of rotors in tilted fields. Energy-level correlation diagrams are computed, and the presence of avoided crossings quantified by the study of nearest-neighbor spacing distributions as a function of energy and tilting angle. Finally, we examine the influence of classical periodic orbits on rotor wave functions. Many wave functions in the tilted field case are found to be highly nonseparable in spherical polar coordinates. Localization of wave functions in the vicinity of classical periodic orbits, both stable and unstable, is observed for many states.

We present several exact solutions of the eigenfrequency problem for torsional shear vibrations in homogeneous and non-homogeneous models of the neutron star crust governed by the canonical equation of solid mechanics with a restoring force of Hookean elasticity. Particular attention is given to the regime of large-lengthscale nodeless axisymmetric differentially rotational oscillations, which are treated in spherical polar coordinates, reflecting the real geometry of the neutron star crust. We highlight the distinction between analytic solutions, computed as a function of the multipole degree of nodeless torsional oscillations and fractional depth of the crust, caused by different boundary conditions imposed on the toroidal field of material displacements. The relevance of the considered models to quasiperiodic oscillations, recently detected during the flare of SGR 1806-20 and SGR 1900+14, is discussed.

For pt.II see Proc. R. Soc. London, Ser. A, vol.384, no.1786, p.89-105 (1982). A procedure for solving the few-particle Schrodinger equation exactly is applied to a model system consisting of two identical particles and a massive third particle. The type of interaction potential is not specified except that it should not diverge more rapidly than r-2 at the particle positions. Allowable interactions include the Coulomb and the harmonic oscillator potentials. The principles are illustrated by reference to the spatially symmetric states of the system. The solution has the form of a multipole expansion in spherical polar coordinates. The radial dependence for each multipole component is defined by a power series including logarithmic terms. Explicit expressions are given for the coefficients in the expansion. Exact coefficients are obtained for low-order terms in the wavefunction of a two-electron atom.

Some useful discrete variable representations for problems in time-dependent and time-independent quantum mechanics

We propose a simple scheme for decomposition of molecular functions into single-center components. The problem of three-dimensional integration in molecular systems thus reduces to a sum of one-center, atomic-like integrations which are treated using standard numerical techniques in spherical polar coordinates. The resulting method is tested on representative diatomic and polyatomic systems for which we obtain five- or six-figure accuracy using a few thousand integration points per atom.

Rotational splitting Δω(n,l,m) of the eigenfrequencies of a star rotating with angular velocity Ω(r, θ) about a unique axis can be represented as a weighted integral of Ω over r and θ, (r, θ, φ) being spherical polar coordinates about the axis of rotation. For high-frequency acoustic modes, Δω/m collapses essentially to a function of ω — ω /(l+1/2) and M = m/(l + 1/2) alone, and the weighting kernel K(r,θ) becomes asymptotically degenerate, each factor being of essentially Abel type. Therefore, formally, the splitting integral can be inverted, once a procedure has been found for extending Δω over the domain of (ω, M) such that the turning points (rt,ωt), given by (c(rt)/rt, sinωt) = (ω, M) where c is sound speed, span the star. Obtaining that representation is the most difficult stage of the inversion. We report on a procedure that treats the inverted two-dimensional Abel integral as a repeated double integral, representing the data successively along a set of parallel lines M =constant. The method is illustrated by an inversion of artificial data which is compared with the angular velocity from which those data were computed.