Numerical solution of unsteady incompressible Navier-Stokes equations using high-order method of lines

A high-order method of lines is devised for solving the unsteady incompressible Navier-Stokes equations in the vorticity-stream function formulation. The vorticity transport equation is solved by the eight- or tenth-order method of lines and the Poisson equation for the stream function is solved by a high-order multigrid method. The numerical results of the two-dimensional (2D) homogeneous isotropic turbulence and the turbulent mixing layer are presented. In the homogeneous isotropic turbulence with tenth order of spatial accuracy, the power law of the inertial energy spectrum at the climax stage coincides with the predictions by Batchelor, Leith and Kraichnan. In the turbulent mixing layer with eight order of spatial accuracy, the vortex pairing are reproduced and the coherent structure of the Reynolds stress at the pairing is noticed.     

Extensional wave motion in homogenous isotropic thermoelastic plate by using asymptotic method

The present investigation is concerned with the study of extensional wave motion in an infinite homogenous isotropic, thermoelastic plate by using asymptotic method. The governing equation for the extensional wave motions have been derived from the system of three-dimensional dynamical equations of linear coupled theory of thermoelasticity. All coefficients of the differential operator are expressed as explicit functions of the material parameters. The velocity dispersion equation for the extensional wave motion is deduced from the three-dimensional analog of Rayleigh–Lamb frequency equation for thermoelastic plate waves. The approximations for long and short waves and expression for group velocity are also derived. The thermoelastic Rayleigh–Lamb frequency equations for the considered plate are expanded in power series in order to obtain polynomial frequency and velocity dispersion relations whose equivalence is established to that of asymptotic method. The dispersion curves for phase velocity and attenuation coefficient are shown graphically for extensional wave motion of the plates.     

A one-dimensional model for dispersive wave turbulence

Summary A family of one-dimensional nonlinear dispersive wave equations is introduced as a model for assessing the validity of weak turbulence theory for random waves in an unambiguous and transparent fashion. These models have an explicitly solvable weak turbulence theory which is developed here, with Kolmogorov-type wave number spectra exhibiting interesting dependence on parameters in the equations. These predictions of weak turbulence theory are compared with numerical solutions with damping and driving that exhibit a statistical inertial scaling range over as much as two decades in wave number. It is established that the quasi-Gaussian random phase hypothesis of weak turbulence theory is an excellent approximation in the numerical statistical steady state. Nevertheless, the predictions of weak turbulence theory fail and yield a much flatter (|k|^−1/3) spectrum compared with the steeper (|k|^−3/4) spectrum observed in the numerical statistical steady state. The reasons for the failure of weak turbulence theory in this context are elucidated here. Finally, an inertial range closure and scaling theory is developed which successfully predicts the inertial range exponents observed in the numerical statistical steady states.     

Asymptotic series in dynamics of fluid flows: Diffusion versus bifurcations

The Brownian motion over the space of fluid velocity configurations driven by the hydrodynamical equations is considered. The Green function is computed in the form of an asymptotic series close to the standard diffusion kernel. The high order asymptotic coefficients are studied. Similarly to the models of quantum field theory, the asymptotic contributions show a factorial growth and are summated by means of Borel’s procedure. The resulting corrected diffusion spectrum has a closed analytical form. The approach provides a possible ground for the optimization of existing numerical simulation algorithms and can be used for the analysis of other asymptotic series in turbulence.     

Spectral solution of free surface flows

A spectral Fourier-Chebyshev method for calculating unsteady two-dimensional free surface flows is presented and discussed. The vorticity-stream function equations are used in association with an influence matrix technique for prescribing the boundary and free surface conditions. The stability of the time-discretization scheme is analysed. Finally, numerical results are given for various physical problems.     

Numerical simulations for the contraction flow using grid generation

We study the incomprssible Navier Stokes equations for the flow inside contraction geometry. The governing equations are expressed in the vorticity-stream function formulations. A rectangular computational domain is arised by elliptic grid generation technique. The numerical solution is based on a technique of automatic numerical generation of acurvilinear coordinate system by transforming the governing equation into computational plane. The transformed equations are approximated using central differences and solved simultaneously by successive over relaxation iteration. The time dependent of the vorticity equation solved by using explicit marching procedure. We will apply the technique on several irregularshapes.     

Die Theorie der asymptotischen Verteilung und die numerische Lösung von Integrodifferentialgleichungen

Summary In this paper terms and theorems of the theory of asymptotic distribution are used investigating numerical methods for the solution of certain integrodifferential equations. These simulation methods are specially applied on the study of collisionless plasmas, which are described by the so called Vlasov-equation. Interpretating the solution of such equations as the asymptotic distribution of suitably constructed point sequences, one gets a mathematical frame for these methods and succeeds in proving convergence.     

Computational organizational network modeling: Strategies and an example

This paper articulates the logic of computational organizational modeling as a strategy for theory construction and testing in the field of organizational communication networks. The paper introduces, Blanche, and objectoriented simulation environment that supports quantitative modeling and analysis of the evolution of organizational networks. Blanche relies on the conceptual primitives of attributes that describe network nodes and links that connect these nodes. Difference equations are used to model the dynamic properties of the network as it changes over time. This paper describes the design of Blanche and how it supports both the process of theory construction, model building and analysis of results. The paper concludes with an empirical example, to test the predictions of a network-based social influence model for the adoption of a new communication technology in the workplace.     

Properties and asymptotic behaviour of the solutions of coupled diffusion equations with time-periodic,space-independent coefficients,with an application to electrodiffusion

Summary We study a Markovian process, the state space of which is the product of a set ofn points and the realx-axids. Under certain regularity conditions this study is equivalent to investigating the solution of a set of couple diffusion equations, generalization of the Fokker-Planck (or second Kolmogorov) equation. Assuming the process homogeneous inx, but in general time-inhomogeneous, this set of equations is studied with the help of the Fourier transformation. The marginal distribution in then discrete states corresponds to a time-inhomogeneousn-state Markov chain in continuous time. The properties of such a Markov chain are studied, especially the asymptotic behaviour in the time-periodic case. We obtain a natural generalization of the well-known asymptotic behaviour in the time-homogeneous case, finding a subdivision of the states into groups of essential states, the distribution inside easch group being asymptotically periodic and independent of the starting distribution. Next, still assuming time-periodicity, we study the asymptotic behaviour of the complete Markovian process, showing that inside each of the groups mentioned above the distribution approaches a common normal distribution inx-space, with mean value and variance proportional tot. Explicit expressions for the proportionality factors are derived. The general theory is applied to the electrodiffusion equations, corresponding ton=2.     

The Asymptotic and Numerical Inversion of the Marcum Q‐Function

The generalized Marcum functions appear in problems of technical and scientific areas such as, for example, radar detection and communications. In mathematical statistics and probability theory these functions are called the noncentral gamma or the noncentral chi‐squared cumulative distribution functions. In this paper, we describe a new asymptotic method for inverting the generalized Marcum Q‐function and for the complementary Marcum P‐function. Also, we show how monotonicity and convexity properties of these functions can be used to find initial values for reliable Newton or secant methods to invert the function. We present details of numerical computations that show the reliability of the asymptotic approximations.     

Breaking of two-dimensional relativistic electron oscillations under small deviations from axial symmetry

A numerical asymptotic model for the breaking of two-dimensional plane relativistic electron oscillations under a small deviation from axial symmetry is developed. The asymptotic theory makes use of the construction of time-uniformly applicable solutions to weakly nonlinear equations. A special finite-difference algorithm on staggered grids is used for numerical simulation. The numerical solutions of axially symmetric one-dimensional relativistic problems yield two-sided estimates for the breaking time. Some of the computations were performed on the “Chebyshev” supercomputer (Moscow State University).     

Extrapolation algorithms for solving nonlinear boundary integral equations by mechanical quadrature methods

We study the numerical solution procedure for two-dimensional Laplace’s equation subjecting to non-linear boundary conditions. Based on the potential theory, the problem can be converted into a nonlinear boundary integral equations. Mechanical quadrature methods are presented for solving the equations, which possess high accuracy order O(h ³) and low computing complexities. Moreover, the algorithms of the mechanical quadrature methods are simple without any integration computation. Harnessing the asymptotical compact theory and Stepleman theorem, an asymptotic expansion of the errors with odd powers is shown. Based on the asymptotic expansion, the h ³ −Richardson extrapolation algorithms are used and the accuracy order is improved to O(h ⁵). The efficiency of the algorithms is illustrated by numerical examples.     

Conservative finite-difference scheme for computer simulation of contrast 3D spatial-temporal structures induced by a laser pulse in a semiconductor

The numerical approach for computer simulation of femtosecond laser pulse interaction with a semiconductor is considered under the formation of 3D contrast time-dependent spatiotemporal structures. The problem is governed by the set of nonlinear partial differential equations describing a semiconductor characteristic evolution and a laser pulse propagation. One of the equations is a Poisson equation concerning electric field potential with Neumann boundary conditions that requires fulfillment of the well-known condition for Neumann problem solvability. The Poisson equation right part depends on free-charged particle concentrations that are governed by nonlinear equations. Therefore, the charge conservation law plays a key role for a finite-difference scheme construction as well as for solvability of the Neumann difference problem. In this connection, the iteration methods for the Poisson equation solution become preferable than using direct methods like the fast Fourier transform. We demonstrate the following: if the finite-difference scheme does not possess the conservatism property, then the problem solvability could be broken, and the numerical solution does not correspond to the differential problem solution. It should be stressed that for providing the computation in a long-time interval, it is crucial to use a numerical method that possessing asymptotic stability property. In this regard, we develop an effective numerical approach—the three-stage iteration process. It has the same economic computing expenses as a widely used split-step method, but, in contrast to the split-step method, our method possesses conservatism and asymptotic stability properties. Computer simulation results are presented.     

Evolution equations associated with non-isothermal phase separation: Subdifferential approach

This paper is concerned with a non-isothermal phase separation model, for instance, describing component separation dynamics for a class of binary systems (solid-solid systems). Our model is a couple of nonlinear parabolicPDE's which are possibly degenerate and singular at the same time. It will be shown that our model can be reformulated as a nonlinear evolution equation governed by subdifferential operator in a Hilbert space, and its mathematically basic questions, such as existence and uniqueness, can be handled in the abstract theory of nonlinear evolution equations in Hilbert spaces. Entrata in Redazione il 15 luglio 1997 e, in versione riveduta, il 25 ottobre 1997.     

On nonlinear neutron transport equations

The purpose of this paper is to investigate a class of time-dependent neutron transport equations in which the total and differential scattering cross sections are nonlinear functions of neutron density function. Sufficient conditions on the nonlinear cross sections are given to insure the existence, uniqueness and asymptotic stability of a solution in one, two, or three-dimensional space domains under various boundary and initial conditions. The approach to the problem is based on abstract analysis on nonlinear evolution equations which are closely related to nonlinear semigroup theory.     

Homogenization Method in the Problem of Long Wave Propagation from a Localized Source in a Basin over an Uneven Bottom

In the framework of the linearized shallow water equations, the homogenization method for wave type equations with rapidly oscillating coefficients that generally cannot be represented as periodic functions of the fast variables is applied to the Cauchy problem for the wave equation describing the evolution of the free surface elevation for long waves propagating in a basin over an uneven bottom. Under certain conditions on the function describing the basin depth, we prove that the solution of the homogenized equation asymptotically approximates the solution of the original equation. Model homogenized wave equations are constructed for several examples of one-dimensional sections of the real ocean bottom profile, and their numerical and asymptotic solutions are compared with numerical solutions of the original equations.     

Approximate analytical solutions of a class of boundary layer equations over nonlinear stretching surface

Third order nonlinear ordinary differential equations, subject to appropriate boundary conditions arising in fluid dynamics, are solved using three different methods viz., the Dirichlet series, method of stretching of variables, and asymptotic function method. Similarity transformations are used to convert the governing partial differential equations into nonlinear ordinary differential equations. The numerical results obtained from the above methods for various problems are given in terms of skin friction. Our study revealed that the results obtained from these methods agree well with those of direct numerical simulation of ordinary differential equations. Also, these methods have advantages over pure numerical methods in obtaining derived quantities such as velocity profile accurately for various values of the parameters at a stretch.     

Interaction of a Solitary Wave with an External Force Moving with Variable Speed

The interaction of a solitary wave with an external force moving with constant acceleration is studied within the forced Korteweg-de Vries equation. For the case of a weak isolated force an asymptotic model based on equations for the amplitude and position of the solitary wave is developed. Phase portraits for this asymptotic system are obtained analytically and numerically. Analysis has shown that an accelerated force of either sign can capture a solitary wave if the acceleration is less than a certain critical value, depending on the forcing amplitude (for the case of a constant force speed only a positive force can capture a solitary wave). Direct numerical simulation of the forced Korteweg-de Vries equation has confirmed the predictions of the asymptotic model. Also, it is shown numerically that the accelerated force can capture more than one solitary wave.     

Asymptotics and numerics of polynomials used in Tricomi and Buchholz expansions of Kummer functions

Expansions in terms of Bessel functions are considered of the Kummer function ₁ F ₁(a; c, z) (or confluent hypergeometric function) as given by Tricomi and Buchholz. The coefficients of these expansions are polynomials in the parameters of the Kummer function and the asymptotic behavior of these polynomials for large degree is given. Tables are given to show the rate of approximation of the asymptotic estimates. The numerical performance of the expansions is discussed together with the numerical stability of recurrence relations to compute the polynomials. The asymptotic character of the expansions is explained for large values of the parameter a of the Kummer function.     

一类具有脉冲免疫治疗的HIV-1感染模型的动力学分析

该文基于一类HIV-1感染免疫治疗模型，研究了一类具有脉冲免疫治疗的HIV-1感染模型.借助脉冲微分方程理论，研究了脉冲免疫治疗模型解的非负性和一致有界性.利用Floquet乘子理论和微分方程的比较定理，推导出脉冲免疫模型无感染周期解局部和全局渐近稳定以及HIV-1一致持续的阈值条件.通过数值模拟，比较了3种不同治疗方案的治疗效果，验证了脉冲免疫治疗的有效性.数值模拟结果表明，当药物输入量足够大或服药间隔适当短时，从理论上可以有效控制甚至根除病毒.