Axis Singularity Research Articles

Axisymmetric Electromagnetic Wave Propagation Computation Using the Constrained Interpolation Profile Scheme With Large Time Steps

Conventional explicit time-domain methods used to solve Maxwell's equations are reliable and robust but are conditionally stable and often require the fulfillment of the condition Courant-Friedrichs-Lewy (CFL) ≤ 1. While this is acceptable for many applications, in some instances, where Maxwell's equations are solved alongside systems with slower propagation velocities, explicit methods prove costly. This is the case for nonrelativistic electromagnetic Particle-In-Cell methods that are required to study plasma thrusters. Several algorithms have been proposed to retain a nearly explicit formulation using large time steps to achieve higher CFL values. Among these is the semi-Lagrangian constrained interpolation profile (CIP) method. While the ability of this method to handle CFL > 1 has been demonstrated for planar 2-D-3-D cases, this has not been done for 2-D cases with cylindrical symmetry. In this article, a procedure is presented to compute the electromagnetic wave propagation in 2-D domains with cylindrical symmetry using the CIP method. The CIP scheme is extended for CFL ≥ 1 cases, and a ghost node method is proposed to deal with the axis singularity and with the wall boundary condition. The results are compared with the fields of a Hertzian dipole and with a coaxial cable, and they show a good agreement.

GAMERA: A Three-dimensional Finite-volume MHD Solver for Non-orthogonal Curvilinear Geometries

Abstract Efficient simulation of plasmas in various contexts often involves the use of meshes that conform to the intrinsic geometry of the system under consideration. We present here a description of a new magnetohydrodynamic code, GAMERA (Grid Agnostic MHD for Extended Research Applications), designed to combine geometric flexibility with high-order spatial reconstruction and constrained transport to maintain the divergence-free magnetic field. GAMERA carries on the legacy of its predecessor, the LFM (Lyon–Fedder–Mobarry), a research code whose use in space physics has spanned three decades. At the time of its initial development, the LFM code had a number of novel features: eighth-order centered spatial differencing, the Partial Donor Cell Method limiter for shock capturing, a non-orthogonal staggered mesh with constrained transport, and conservative averaging-reconstruction for axis singularities. The capability to handle multiple ion species was also added later. GAMERA preserves the core numerical philosophy of LFM while also incorporating numerous algorithmic and computational improvements. The upgrades in the numerical schemes include accurate grid metric calculations using high-order Gaussian quadrature techniques, high-order upwind reconstruction, non-clipping options for interface values, and improved treatment of axis singularities. The improvements in the code implementation include the use of data structures and memory access patterns conducive to aligned vector operations and the implementation of hybrid parallelism, using MPI and OMP. GAMERA is designed to be a portable and easy-to-use code that implements multidimensional MHD simulations in arbitrary non-orthogonal curvilinear geometries on modern supercomputer architectures.

Conservative averaging-reconstruction techniques (Ring Average) for 3-D finite-volume MHD solvers with axis singularity

In cylindrical/spherical geometries, MHD solvers using explicit finite-volume methods face restrictive time steps imposed by the CFL condition due to the clustering of cells in the azimuthal direction near the pole/axis. We use a conservative averaging-reconstruction method (Ring Average) on structured cylindrical/spherical mesh to remove this severe time step restriction for multi-dimensional curvilinear finite-volume MHD solvers. The Ring Average technique is implemented as a post-processing step and thus requires no changes to the existing data structure, grid definition or numerical methods in the original MHD solver. This paper describes the Ring Average algorithm and presents simulation results using field loop advection and MHD blast waves in cylindrical/spherical geometries for validation. The algorithm is shown to be inexpensive and easily implemented in existing curvilinear finite-volume MHD solvers.

Stable axisymmetric SPH formulation with no axis singularity

SummaryThe axisymmetric formulation of the governing equations for geomechanics in the framework of smoothed particle hydrodynamics (SPH) is presented in this study. Two forms of SPH discretization for the motion equations, which are labeled as form I and form II, are proposed, and the methods to compute the hoop stress and strain terms including hoop strain rate and the acceleration introduced by the hoop stress are compared. To avoid possible singularity problem near the axis of symmetry, a perfectly smooth contact along with ghost particles are applied to prevent the real particles from overly approaching the axis of symmetry to remove this potential singularity. In addition, the Mohr–Coulomb constitutive model is implemented into the SPH formulation in describing soil behavior. Four numerical tests are carried out to validate and compare the accuracy and stability of the proposed algorithms, and their results are compared with analytical solutions and results from FEM analysis. The performance in these comparisons suggests that SPH II with hoop terms computed through direct hoop method is more stable than the others, and the adoption of contact for the symmetric axis is efficient in eliminating the singularity problem. Copyright © 2015 John Wiley & Sons, Ltd.

Wave Systems with an Infinite Number of Localized Traveling Waves

In many wave systems, propagation of steadily traveling solitons or kinks is prohibited because of resonances with linear excitations. We show that wave systems with resonances may admit an infinite number of traveling solitons or kinks if the closest to the real axis singularities of a limiting asymptotic solution in the complex upper half plane are of the form z±=±α+iβ, α≠0. This quite general statement is illustrated by examples of the fifth-order Korteweg-de Vries equation, the discrete cubic-quintic Klein-Gordon equation, and the nonlocal double sine-Gordon equations.

On analytic properties of periodic solutions for equation

The paper focuses on 2L-periodic solutions, UL(x), of the nonlocal equation (p > 2 is integer, is the Hilbert transform). We give numerical evidence for the existence of a continuous family of these solutions parametrized by L for p = 3, 4, 5. The features of the solutions UL(x) are discussed. In particular, we offer strong numerical arguments that UL(x) can be continued analytically from the real axis to some strip in the complex plane, UL(x) → UL(z). The singularities which arise under analytical continuation of these solutions into the complex plane are considered. It is proved (theorem 1) that UL(z) cannot be any power of a meromorphic function. The formula which describes the asymptotic behavior of UL(z) in a neighborhood of the singularity is given. It agrees with the numerical results; computations also allow us to locate the closest to the real axis singularity of UL(z) and estimate the width of the analyticity strip.

Two-dimensional hydrodynamic core-collapse supernova simulations with spectral neutrino transport

Spherically symmetric (ID) and two-dimensional (2D) supernova simulations for progenitor stars between 11 M ⊙ and 25 M ⊙ are presented, making use of the PROMETHEUS/VERTEX neutrino-hydrodynamics code, which employs a full spectral treatment of neutrino transport and neutrino-matter interactions with a variable Eddington factor closure of the O(v/c) moments equations of neutrino number, energy, and momentum. Multi-dimensional transport aspects are treated by the ray-by-ray plus approximation described in Paper I. We discuss in detail the variation of the supernova evolution with the progenitor models, including one calculation for a 15 M ⊙ progenitor whose iron core is assumed to rotate rigidly with an angular frequency of 0.5 rad s -1 before collapse. We also test the sensitivity of our 2D calculations to the angular grid resolution, the lateral wedge size of the computational domain, and to the perturbations which seed convective instabilities in the post-bounce core. In particular, we do not find any important differences depending on whether random perturbations are included already during core collapse or whether such perturbations are imposed on a ID collapse model shortly after core bounce. Convection below the neutrinosphere sets in 30-40 ms after bounce at a density well above 10 12 g cm -3 in all 2D models, and encompasses a layer of growing mass as time goes on. It leads to a more extended proto-neutron star structure with reduced mean energies of the radiated neutrinos, but accelerated lepton number and energy loss and significantly higher muon and tau neutrino luminosities at times later than about 100ms after bounce. While convection inside the nascent neutron star turns out to be insensitive to our variations of the angular cell and grid size, the convective activity in the neutrino-heated postshock layer gains more strength in better resolved models. We find that low (l = 1, 2) convective modes, which require the use of a full 180 degree grid and are excluded in simulations with smaller angular wedges, can qualitatively change the evolution of a model. In case of an 11.2 M ⊙ star, the lowest-mass progenitor we investigate, a probably rather weak explosion by the convectively supported neutrino-heating mechanism develops after about 150ms post-bounce evolution in a 2D simulation with 180 degrees, whereas the same model with 90 degree wedge fails to explode like all other models. This sensitivity demonstrates the proximity of our 2D calculations to the borderline between success and failure, and stresses the need to strive for simulations in 3D, ultimately without the constraints connected with the axis singularity of a polar coordinate grid.

On complex singularities of solutions of the equation ℋux-u+up= 0

It has been proved recently by Bona and Li Yi (Bona J L and Li Yi A 1997 J. Math. Pure Appl. 76 377) that solitary wave solutions of a certain class of nonlinear nonlocal equations can be extended into the complex domain. In this paper we present an approach for the study of singularities of such extensions. This approach is applied to the localized solutions of the equation Hux − u + u p = 0, p = 3, 4, 5 where H is the Hilbert transform. The location of the closest to real axis singularity z = z0 was found numerically; the analysis of this type of singularity shows that in the vicinity of z = z0 these complex extensions of solutions cannot be represented by the series 1 (z−z0)ρ � ∞ n=0 An(z − z0) n , i.e. they do not correspond to some power of a meromorphic function.

Direct Numerical Simulation of the Puffing Phenomenon of an Axisymmetric Thermal Plume

A spatial direct numerical simulation of an axisymmetric buoyant thermal plume is presented. The governing flow field equations at the centerline are put into a special form to circumvent the axis singularity associated with the cylindrical coordinates and the high order accuracy of the numerical scheme is preserved at the centerline. Boundary conditions associated with the spatial DNS of open-boundary buoyant flows and compatible with the modern nondissipative high-order finite difference schemes have been developed. The fluid exhibits a periodic oscillatory motion known as the puffing phenomenon, which is the formation and convection of vortex at the near field of the plume. Budgets of the vorticity transport are determined to examine the mechanisms leading to the puffing phenomenon. The analysis on vorticity transport shows that vorticity is created mainly by the gravitational term which is due to the interaction between the radial density gradients and gravity at the initial stage of the establishment of the puffing structure, while the baroclinic torque dominates the vorticity transport when the flow is established. Density stratification in the radial direction close to the plume base is found to be essential to the development of the buoyant flow instability. Simulations with different initial temperature ratios reveal that entrainment close to the plume base is enhanced at a higher temperature ratio despite the fact that the puffing structures and the plume pulsation frequency only vary very weakly with the initial temperature ratio. The predicted puffing frequencies are in agreement with the values from experimental correlations for fire and isothermal helium/air plumes.

An Efficient Method for the Solution of the Incompressible Navier–Stokes Equations in Cylindrical Geometries

The article presents a fast pseudo-spectral Navier–Stokes solver for cylindrical geometries, which is shown to possess exponential rate of decay of the error. The formulation overcomes the issues related to the axis singularity, by employing in the radial direction a special set of collocation points together with standard Chebyshev polynomials. A multi-domain technique with patching interfaces yields significant improvements in the conditioning of the algebraic problems arising from the discretization procedure and allows for an enhanced near wall resolution of wall bounded shear flows. The elliptic kernel enjoys the efficiency of an analytic expansion of the harmonic extension. The method is tested by computing the formation of Taylor vortices in a rotating Couette flow for both axisymmetric and non-axisymmetric configurations. A direct numerical simulation of a turbulent pipe flow at moderate Reynolds number demonstrates the effectiveness of the method in as much as the axis singularity is concerned. Results compare well with reference experimental and numerical data.

Effective treatment of the singular line boundary problem for three-dimensional grids

The axis singularity problem for 3D blunt body flows is examined. Two techniques of eliminating the axis-singularity difficulty are presented for utilization with finite-difference codes: (1) the 3D Navier-Stokes equations are reformulated utilizing a redefined Jacobian that is nonzero at the singular line, and (2) the 3D Navier-Stokes equations are solved using the Roe flux-difference splitting technique with a finite-volume based method for evaluation of the grid Jacobian and the metrics, and appropriate boundary conditions. Real-gas and viscous computations are also presented, demonstrating the generality of the two methods.

Saffman-Taylor fingers in the sector geometry.

We study analytically the selection, due to surface tension, of Saffman-Taylor fingers in the sector geometry recently introduced experimentally by Thom\'e et al. [Phys. Fluids A 1, 224 (1989)]. The finger shape for zero surface tension is written as the solution of an integral equation. This leads to an explicit solution in the case where the sector angle is small. The selection due to surface tension is controlled, as in similar problems, by the singular points of a differential equation. For divergent fingers, we find for these singularities a geometric situation which is qualitatively new. Indeed we obtain one ``axis'' singularity and two ``side'' singularities. When the axis singularity is ``dominant,'' the finger disappears, which explains the branch-merging phenomenon found numerically by Ben Amar [Phys. Rev. A 44, 3673 (1991)]. Our analysis is carried out completely explicitly in the case of small-angle sector, and we indeed find branch merging, either by a WKB analysis or by a numerical solution of our differential equation. For convergent fingers, we find either an ``axis'' singularity or two ``side'' singularities, and the results are similar to those for the standard Saffman-Taylor fingers in the linear geometry.

Off the Real Axis Singularities in Partial-Wave Amplitudes

Off the Real Axis Singularities in Partial-Wave Amplitudes

Notes on bridged-T complex conjugate compensation and 4-terminal network loading

MANY servo systems possess open-loop transfer functions which contain one or more pairs of complex conjugate poles. Often a pair of these complex conjugate poles will lie close to the imaginary axis and will cause the uncompensated system to possess a relatively poor damping ratio or damping exponent. Usually, satisfactory compensation of such systems can be achieved through the use of tachometer feedback or the conventional compensators possessing real axis singularities. However, there are occasions when complex conjugate singularity compensation would possess distinct advantages. This is particularly true if the compensating complex singularities can be introduced by a simple R-C (resistance-capacitance) passive network with zero d-c attenuation.