Higher-order Derivative Terms Research Articles

A smoothing algorithm for the higher-order derivative terms in a nonlinear k-ε turbulence model

The nonlinear k-e turbulence models are capable of accurately predicting the Reynolds stresses in recirculating flows with strong anisotropy as opposed to standard or linear k-e model. However, the nonlinear models introduce higher-order derivatives of the mean velocity field and their products in the stress tensor. The inherent oscillations of the higher-order nonlinear difference equations need to be eliminated which can be achieved if the velocity field is smoothed out before calculating the derivatives. Otherwise, the computation procedure quickly becomes unstable. In this study, a new localised smoothing filter for the velocity field, based on the least squares method is proposed and evaluated using the nonlinear k-e model and the widely used bench mark test case of the backward facing step problem. The smoothing algorithm effectively stabilised the computation of the nonlinear k-e model with a modest increase in CPU time compared to the standard k-e model computation.

Undular bores and secondary waves -Experiments and hybrid finite-volume modelling

Secondary free-surface undulations (Favre waves), appearing for example after the opening of a sluice gate or at the head of a bore, cannot be reproduced by numerical models based on the hydrostatic pressure assumption. The Boussinesq equations take into account the extra pressure gradients but are difficult to integrate due to the high-order derivative terms. The paper describes the physics of wave initiation and proposes a demonstration of the Boussinesq equation based on relatively wider assumptions than usually adopted. A linear stability analysis is developed in finite-difference frame to highlight some potential source of numerical instabilities. These conclusions are transposed in a new hybrid finite-volume / finite-difference scheme, which reveals a better accuracy in period and amplitude when evaluated against experiments.

Periodic vacuum and particles in two dimensions

Different dynamical symmetry breaking patterns are explored for the two dimensional phi4 model with higher order derivative terms. The one-loop saddle point expansion predicts a rather involved phase structure and a new Gaussian critical line. This vacuum structure is corroborated by the Monte Carlo method, as well. Analogies with the structure of solids, the density wave phases and the effects of the quenched impurities are mentioned. The unitarity of the time evolution operator in real time is established by means of the reflection positivity.

On the equivalence between noncommutative and ordinary gauge theories

Recently Seiberg and Witten have proposed that noncommutative gauge theories realized as effective theories on D-brane are equivalent to some ordinary gauge theories. This proposal has been proved, however, only for the Dirac-Born-Infeld action in the approximation of neglecting all derivative terms. In this paper we explicitly construct general forms of the 2n-derivative terms which satisfy this equivalence under their assumption in the approximation of neglecting (2 n+2)-derivative terms. We also prove that the D-brane action computed in the superstring theory is consistent with the equivalence neglecting the fourth and higher order derivative terms.

Application of mapping closure to non-Gaussian velocity fields

An attempt is made to assess the mapping closure for turbulent velocity fields with non-Gaussian probability density functions (PDFs). The a priori assumption of the Gaussian velocity statistics is not made in the present mapping closure, which generalizes the results for the Burgers turbulence by Kraichnan [Phys. Rev. Lett. 65, 575 (1990)] and Gotoh and Kraichnan [Phys. Fluids A 5, 445 (1993)]. The PDFs calculated from the mapping closure are compared to the results obtained from the direct numerical simulation for non-Gaussian initial fields. The statistics of second- and higher-order derivative terms are discussed also.

Higher-order black-hole solutions in N=2 supergravity and Calabi-Yau string backgrounds

Abstract Based on special geometry, we consider corrections to N=2 extremal black-hole solutions and their entropies originating from higher-order derivative terms in N=2 supergravity. These corrections are described by a holomorphic function, and the higher-order black-hole solutions can be expressed in terms of symplectic Sp(2n+2) vectors. We apply the formalism to N=2 type-IIA Calabi-Yau string compactifications and compare our results to recent related results in the literature.

Feedback control of congestion in packet switching networks: the case of multiple congested nodes

In this paper, an analytical method for the design of a congestion control scheme in packet switching networks is presented. This scheme is particularly suitable for implementation in ATM switches, for the support of the available bit rate (ABR) service in ATM networks. The control architecture is rate-based with a local feedback controller associated with each switching node. The controller itself is a generalization of the standard proportional-plus-derivative controller, with the difference that extra higher-order derivative terms are involved to accommodate the delay present in high-speed networks. It is shown that, under the specific service discipline introduced here, there exists a set of control gains that result in asymptotic stability of the linearized network model. A method for calculating these gains is given. In addition, it is shown that the resulting steady state rate allocation possesses the so-called max-min fairness property. The theoretical results are illustrated by a simulation example, where it is shown that the controller designed, using the methods developed here, works well for both the service discipline introduced in this paper and for the standard FCFS scheme. © 1997 John Wiley & Sons, Ltd.

Duality in N = 1, D = 10 superspace and supergravity with tree level superstring corrections

The equivalence of usual (with the 3-form graviphoton field) and dual (with the 7-form graviphoton field) formulation of N = 1, D = 10 supergravity is established on the mass shell with taking into account high order derivative terms which are necessary for cancellation of anomalies. The bosonic part of the dual supergravity Lagrangian with the Green-Schwarz anomaly compensating term and all other terms connected by the supersymmetry is obtained explicitely for the first time.

Chiral Schwinger model with a Podolsky term at finite temperature

We study an exactly solvable two-dimensional model which mimics the basic features of the standard model. This model combines chiral coupling with an infrared behavior which resembles low energy QCD. This is done by adding a Podolsky higher-order derivative term in the gauge field to the Lagrangian of the usual chiral Schwinger model. We adopt a finite temperature regularization procedure in order to calculate the non-trivial fermionic Jacobian and obtain the photon and fermion propagators, first at zero temperature and then at finite temperature in the imaginary and real time formalisms. Both singular and non-singular cases, corresponding to the choice of the regularization parameter, are treated. In the nonsingular case there is a tachyonic mode as usual in a higher order derivative theory, however in the singular case there is no tachyonic excitation in the spectrum.

The Effect of Friction on the Forward Dynamics Problem

This article discusses the numerical solution of the forward dynamic equations of an n-degree-of-freedom manipulator with friction. Also discussed are the modeling and experimen tal identification of friction. It is shown that the inclusion of Coulomb-type friction in the dynamic equations introduces two difficulties in the forward dynamic solution. The differential equations are shown to be discontinuous in the highest-order derivative terms. In addition, the load dependency of this type of friction typically causes the equations to be implicit in the joint accelerations. For the important case of load-dependent transmission friction, the equations can be explicit. Techniques for the forward solution are described through the example of a roller screw transmission. Experimental and simulation results are used to show the importance of load-dependent friction in a particular robot.

Radiation by solitons due to higher-order dispersion.

We consider the Korteweg--de Vries (KdV) and nonlinear Schr\"odinger (NS) equations with higher-order derivative terms describing dispersive corrections. Conditions of existence of stationary and radiating solitons of the fifth-order KdV equation are obtained. An asymptotic time-dependent solution to the latter equation, describing the soliton radiation, is found. The radiation train may be in front as well as behind the soliton, depending on the sign of dispersion. The change rate of the soliton due to the radiation is calculated. A modification of the WKB method, that permits one to describe in a simple and general way the radiation of KdV and NS, as well as other types of solitons, is developed. From the WKB approach it follows that the soliton radiation is a result of a tunneling transformation of the nonlinearly self-trapped wave into the free-propagating radiation.

Influence of high-order dispersion on self-focusing. I. Qualitative investigation

The self-focusing, described by the nonlinear Schrödinger equation with a higher-order derivative term, appearing from dispersive corrections, is considered. A qualitative investigation shows that this term, even if it is small, may play an important role in the final state of the self-focusing. Depending on the sign of a coefficient before this term, it may lead either to a tunneling of the self-trapped radiation, which finally results in defocusing, or to a steady homogeneous wave beam.

Influence of high-order dispersion on self-focusing. II. Numerical investigation

The self-focusing described by the nonlinear Schrödinger equation with a higher-order derivative term is investigated numerically. We demonstrate that, depending on the sign before this term, it may lead in the final stage either to defocusing or to a steady homogeneous wave beam. In both cases the transition to the final state is shown to be oscillatory.

Canonical quantization of the massive vector supermultiplet: an example of higher-order derivative model

The supersymmetric version of the Stueckelberg Lagrangian of the massive vector superfield leads to an example of higher-order derivative model. The canonical quantization yields massive states which compose two irreducible representations, one physical supermultiplet and another spurious. The origin of the different spurious states are investigated, especially those originating from the higher-order derivative terms. The spurious superfield is found to be decoupled when the supercurrent satisfies some appropriate conditions.

Phenomenological implications of a generalized Skyrme model.

We examine a modified SU(2) Skyrme model and its predictions. The model is constructed of higher-order terms in derivatives of the pion field in such a way that the chiral-angle equation remains a differential equation of degree 2.

The Skyrme model and higher order terms

We generalize the Skyrme lagrangian by considering a set of higher order terms in derivatives of the pion field (up to order twelve) using a systematic approach. We then consider a combination of those terms which preserves the equation for the chiral angle as a second-degree differential equation. A repetitive pattern shows up order-by-order suggesting that it could be generalized and perhaps summed to all orders.

Dynamics of the semiclassical Einstein equations.

An investigation is done on the behavior of the Einstein equation for the case of a conformally invariant field in a conformally flat spacetime when higher-order derivative terms with logarithmic dependence on the scalar curvature are introduced. It is seen that in the quantum case, flat spacetime is always stable to conformally flat perturbations.

Three-dimensional eddy current problems using the T-Ω method and fredholm integral equations

A technique is discussed for computing three-dimensional magnetic fields created by an oscillating source in a conducting medium. The technique's contribution lies in the manner in which the Fredholm integral technique is merged with a representation of the magnetic field as the sum of a vector and the gradient of a scalar (the T-Ω representation). The T-Ω representation is shown to conveniently realize the boundary conditions without introducing higher order derivative terms into the integral equations. The technique offers a powerful means of numerically calculating three-dimensional fields, necessitating only the discretization of interfacial surfaces.

Gauss type quadrature rules for Cauchy principal value integrals

Two quadrature rules for the approximate evaluation of Cauchy principal value integrals, with nodes at the zeros of appropriate orthogonal polynomials, are discussed. An expression for the truncation error, in terms of higher order derivatives, is given for each rule. In addition, two theorems, containing sufficient conditions for the convergence of the sequence of quadrature rules to the integral, are proved.

Dynamics of Einstein's equation modified by a higher-order derivative term

The search for a renormalized stress-energy operator of a quantum field in curved spacetime has raised the question of whether one can add terms to Einstein's equation containing fourth-order derivatives of the metric, and still maintain a reasonable theory. We investigate this question by considering the simple case of a conformally invariant field in a conformally flat spacetime, where one has a well-defined prediction for the vacuum stress energy of the field. We find that if a certain fourth-order term (associated with the D/sup 7/AlembertianR trace anomaly) enters with one sign, flat spacetime is unstable to conformally flat perturbations that grow exponentially on the Planck time scale. If this term enters with the other sign, conformally flat perturbations could cause spacetime to oscillate at the Planck frequency, resulting in high-energy radiation by charged test particles.