Numerical studies on quasilinear and linear elliptic equations

We explore here the acceleration of convergence of iterative methods for the solution of a class of quasilinear and linear algebraic equations. The specific systems are the finite difference form of the Navier-Stokes equations and the energy equation for recirculating flows. The acceleration procedures considered are: the successive over relaxation scheme; several implicit methods; and a second-order procedure. A new implicit method—the alternating direction line iterative method—is proposed in this paper. The method combines the advantages of the line successive over relaxation and alternating direction implicit methods. The various methods are tested for their computational economy and accuracy on a typical recirculating flow situation. The numerical experiments show that the alternating direction line iterative method is the most economical method of solving the Navier-Stokes equations for all Reynolds numbers in the laminar regime. The usual ADI method is shown to be not so attractive for large Reynolds numbers because of the loss of diagonal dominance. This loss can however be restored by a suitable choice of the relaxation parameter, but at the cost of accuracy. The accuracy of the new procedure is comparable to that of the well-tested successive overrelaxation method and to the available results in the literature. The second-order procedure turns out to be the most efficient method for the solution of the linear energy equation.

Nature of dependence of spin-orbit splittings on atomic number

It has been shown that Dirac equation employing a constant value of the screening constant Z0 does not explain the variation of spin-orbit splittings of 2p and 3p levels with atomic number Z. A model which takes into account the variation of Z0 withZ is shown to satisfactorily predict the dependence of spinorbit splittings onZ.

Application of the Conjugate Gradient Method to a Problem on Minimum Time Orbit Transfer

This correspondence considers the problem of optimally controlling the thrust steering angle of an ion-propelled spaceship so as to effect a minimum time coplanar orbit transfer from the mean orbital distance of Earth to mean Martian and Venusian orbital distances. This problem has been modelled as a free terminal time-optimal control problem with unbounded control variable and with state variable equality constraints at the final time. The problem has been solved by the penalty function approach, using the conjugate gradient algorithm. In general, the optimal solution shows a significant departure from earlier work. In particular, the optimal control in the case of Earth-Mars orbit transfer, during the initial phase of the spaceship's flight, is found to be negative, resulting in the motion of the spaceship within the Earth's orbit for a significant fraction of the total optimized orbit transfer time. Such a feature exhibited by the optimal solution has not been reported at all by earlier investigators of this problem.

Orbit perturbation theory of beam-plasma instability

The nonlinear theory of the instability caused by an electron beam-plasma interaction is studied. A nonlinear analysis has been carried out using many-body methods. A general formula for a neutral collisionless plasma, without external fields, is derived. This could be used for calculating the saturation levels of other instabilities. The effect of orbit perturbation theory on the beam-plasma instability is briefly reviewed.

Third post-Newtonian angular momentum flux and the secular evolution of orbital elements for inspiralling compact binaries in quasi-elliptical orbits

The angular-momentum flux from an inspiralling binary system of compact objects moving in quasi-elliptical orbits is computed at the third post-Newtonian (3PN) order using the multipolar post-Minkowskian wave generation formalism. The 3PN angular-momentum flux involves the instantaneous, tail, and tail-of-tails contributions as for the 3PN energy flux, and in addition a contribution due to nonlinear memory. We average the angular-momentum flux over the binary's orbit using the 3PN quasi-Keplerian representation of elliptical orbits. The averaged angular-momentum flux provides the final input needed for gravitational-wave phasing of binaries moving in quasi-elliptical orbits. We obtain the evolution of orbital elements under 3PN gravitational radiation reaction in the quasi-elliptic case. For small eccentricities, we give simpler limiting expressions relevant for phasing up to order e(2). This work is important for the construction of templates for quasi-eccentric binaries, and for the comparison of post-Newtonian results with the numerical relativity simulations of the plunge and merger of eccentric binaries.

PIV studies of large scale structures in the near field of small aspect ratio elliptic jets

The near flow field of small aspect ratio elliptic turbulent free jets (issuing from nozzle and orifice) was experimentally studied using a 2D PIV. Two point velocity correlations in these jets revealed the extent and orientation of the large scale structures in the major and minor planes. The spatial filtering of the instantaneous velocity field using Gaussian convolution kernel shows that while a single large vortex ring circumscribing the jet seems to be present at the exit of nozzle, the orifice jet exhibited a number of smaller vortex ring pairs close to jet exit. The smaller length scale observed in the case of the orifice jet is representative of the smaller azimuthal vortex rings that generate axial vortex field as they are convected. This results in the axis-switching in the case of orifice jet and may have a mechanism different from the self induction process as observed in the case of contoured nozzle jet flow.

Some nonstandard error analysis of discontinuous Galerkin methods for elliptic problems

An a priori error analysis of discontinuous Galerkin methods for a general elliptic problem is derived under a mild elliptic regularity assumption on the solution. This is accomplished by using some techniques from a posteriori error analysis. The model problem is assumed to satisfy a GAyenrding type inequality. Optimal order L (2) norm a priori error estimates are derived for an adjoint consistent interior penalty method.

Elliptic flow of thermal photons in heavy-ion collisions at Relativistic Heavy Ion Collider and Large Hadron Collider

We calculate the thermal photon transverse momentum spectra and elliptic flow in $\sqrt{s_{NN}} = 200$ GeV Au+Au collisions at RHIC and in $\sqrt{s_{NN}} = 2.76$ TeV Pb+Pb collisions at the LHC, using an ideal-hydrodynamical framework which is constrained by the measured hadron spectra at RHIC and LHC. The sensitivity of the results to the QCD-matter equation of state and to the photon emission rates is studied, and the photon $v_2$ is discussed in the light of the photonic $p_T$ spectrum measured by the PHENIX Collaboration. In particular, we make a prediction for the thermal photon $p_T$ spectra and elliptic flow for the current LHC Pb+Pb collisions.

Free convection about cylinders of elliptic cross section embedded in a porous medium

Numerical solutions are presented for the free convection boundary layers over cylinders of elliptic cross section embedded in a fluid-saturated porous medium. The transformed conservation equations of the nonsimilar boundary layers are solved numerically by an efficient finite-difference method. The theory was applied to a number of cylinders and the results compared very well with published analytical solutions. The results are of use in the design of underground electrical cables, power plant steam, and water distribution lines, among others.

Peristaltic transport of a biofluid in a pipe of elliptic cross section

Peristaltic transport of two fluids occupying the peripheral layer and the core in an elliptic tube is, investigated in elliptic cylindrical co-ordinate system, under long wavelength and low Reynolds number approximations. The effect of peripheral-layer viscosity on the flow rate and the frictional force for a slightly elliptic tube is discussed. The limiting results for the one-fluid model are obtained for different eccentricities of the undisturbed tube cross sections with the same area. As a result of non-uniformity of the peristaltic wave, two different amplitude ratios are defined and the time-averaged flux and mechanical efficiency are studied for different eccentricities. It is observed that the time-averaged flux is not affected significantly by the pressure drop when the eccentricity is large. For the peristaltic waves with same area variation, the pumping seems to improve with the eccentricity.

Elliptic slot antenna for broadband wireless communications

This paper presents the design of a broadband antenna suitable for wireless communications operating over the frequency range of 3.1-10.6 GHz. Parametric studies on the effect of stub and elliptic slot have been carried out to arrive at optimum dimensions to achieve enhanced bandwidth of the proposed antenna. An experimental antenna has been designed and tested to validate the proposed design. Measured return loss characteristics have been compared against the simulation results. Simulated radiation patterns at 3.1 GHz, 6.85 GHz and 10.6 GHz have also been presented in this paper.

Two-Parameter Uniformly Elliptic Sturm–Liouville Problems With Eigenparameter-Dependent Boundary Conditions

We consider the two-parameter Sturm–Liouville system $$ -y_1''+q_1y_1=(\lambda r_{11}+\mu r_{12})y_1\quad\text{on }[0,1], $$ with the boundary conditions $$ \frac{y_1'(0)}{y_1(0)}=\cot\alpha_1\quad\text{and}\quad\frac{y_1'(1)}{y_1(1)}=\frac{a_1\lambda+b_1}{c_1\lambda+d_1}, $$ and $$ -y_2''+q_2y_2=(\lambda r_{21}+\mu r_{22})y_2\quad\text{on }[0,1], $$ with the boundary conditions $$ \frac{y_2'(0)}{y_2(0)} =\cot\alpha_2\quad\text{and}\quad\frac{y_2'(1)}{y_2(1)}=\frac{a_2\mu+b_2}{c_2\mu+d_2}, $$ subject to the uniform-left-definite and uniform-ellipticity conditions; where $q_{i}$ and $r_{ij}$ are continuous real valued functions on $[0,1]$, the angle $\alpha_{i}$ is in $[0,\pi)$ and $a_{i}$, $b_{i}$, $c_{i}$, $d_{i}$ are real numbers with $\delta_{i}=a_{i}d_{i}-b_{i}c_{i}>0$ and $c_{i}\neq0$ for $i,j=1,2$. Results are given on asymptotics, oscillation of eigenfunctions and location of eigenvalues.

An interior penalty method for a sixth-order elliptic equation

We derive and study a C(0) interior penalty method for a sixth-order elliptic equation on polygonal domains. The method uses the cubic Lagrange finite-element space, which is simple to implement and is readily available in commercial software. After introducing some notation and preliminary results, we provide a detailed derivation of the method. We then prove the well-posedness of the method as well as derive quasi-optimal error estimates in the energy norm. The proof is based on replacing Galerkin orthogonality with a posteriori analysis techniques. Using this approach, we are able to obtain a Cea-like lemma with minimal regularity assumptions on the solution. Numerical experiments are presented that support the theoretical findings.