Showing papers on "Asymptotic analysis published in 2005"
TL;DR: In this paper, the authors apply the asymptotic analysis directly to the fully discrete Boltzmann equation, as opposed to the usual practice of analyzing a continuous equation obtained through the Taylor-expansion of the LBE.
369 citations
TL;DR: The paper provides a general method to derive a channel model which is consistent with one's state of knowledge and useful both in terms of designing a system based on criteria such as quality of service and in optimizing transmissions in multiuser networks.
Abstract: We devise theoretical grounds for constructing channel models for multiple-input multiple-output (MIMO) systems based on information-theoretic tools. The paper provides a general method to derive a channel model which is consistent with one's state of knowledge. The framework we give here has already been fruitfully explored with success in the context of Bayesian spectrum analysis and parameter estimation. For each channel model, we conduct an asymptotic analysis (in the number of antennas) of the achievable transmission rate using tools from random matrix theory. A central limit theorem is provided on the asymptotic behavior of the mutual information and validated in the finite case by simulations. The results are useful both in terms of designing a system based on criteria such as quality of service and in optimizing transmissions in multiuser networks.
194 citations
01 Jan 2005
TL;DR: In this paper, the laminar flamelet equations were extended for a two-feed system with one fuel and one oxidizer stream, which can describe the transfer of heat and mass between two mixture fields in flamelet space.
Abstract: The laminar flamelet equations, which were originally derived for a two-feed system with one fuel and one oxidizer stream, are extended for a three-feed system with two fuel and one oxidizer streams. Both fuel streams are associated with one separate mixture fraction. Using a three-scale asymptotic analysis, two-dimensional flamelet equations are derived, which can describe the transfer of heat and mass between two mixture fields in flamelet space. The representative interactive flamelet model which was previously used successfully for the simulation of Diesel engine combustion cases, is extended to accommodate the two-dimensional flamelet equations and multiple mixture fractions. This new model is applied to a split injection case with a pilot and a main injection representing the two fuel streams. The three-dimensional mixture field in the engine is analyzed using a multi-dimensional β-PDF, and an interaction coefficient is defined to describe the degree of merging of the mixture fields. Depending on the coefficient, different phases of combustion and interaction between the mixture fields resulting from the different injections are identified. Results using the two-dimensional flamelet model are compared to experimental data for a modern direct-injection Diesel engine equipped with a Common-Rail injection system. Possible simplified models are introduced, and their performance is compared to the full model.
169 citations
TL;DR: In this article, the stability of a horizontal fluid saturated sparsely packed porous layer heated from below and cooled form above when the solid and fluid phases are not in local thermal equilibrium is examined analytically.
98 citations
TL;DR: In this article, an efficient algorithm based on a moment-method formulation is presented for the evaluation of the field produced by a line source at the interface between an air superstrate and a one-dimensional periodic artificial material slab.
Abstract: An efficient algorithm based on a moment-method formulation is presented for the evaluation of the field produced by a line source at the interface between an air superstrate and a one-dimensional-periodic artificial-material slab. The formulation provides physical insight into the nature of the fields via path deformation in the complex wavenumber plane. From an asymptotic analysis in the complex wavenumber plane it is found that the space wave produced by a line source consists of an infinite number of space harmonics that decay algebraically as x/sup -3/2/. Guided modes may also exist and be excited, including leaky modes.
84 citations
TL;DR: New asymptotic analysis methods that enable phase field equations to emulate the sharp interface limit even in the case of quite diffuse phase-field interfaces are discussed, an aspect that greatly reduces computation times.
Abstract: We review the use of phase field methods in solidification modeling, describing their fundamental connection to the physics of phase transformations. The inherent challenges associated with simulating phase field models across multiple length and time scales are discussed, as well as how these challenges have been addressed in recent years. Specifically, we discuss new asymptotic analysis methods that enable phase field equations to emulate the sharp interface limit even in the case of quite diffuse phase-field interfaces, an aspect that greatly reduces computation times. We then review recent dynamic adaptive mesh refinement algorithms that have enabled a dramatic increase in the scale of microstructures that can be simulated using phase-field models, at significantly reduced simulation times. Combined with new methods of asymptotic analysis, the adaptive mesh approach provides a truly multi-scale capability for simulating solidification microstructures from nanometers up to centimeters. Finally, we present recent results on 2D and 3D dendritic growth and dendritic spacing selection, which have been made using phase-field models solved with adaptive mesh refinement.
84 citations
TL;DR: In this article, the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle is addressed.
Abstract: The aim of the topological asymptotic analysis is to provide an asymptotic expansion of a shape functional with respect to the size of a small inclusion inserted inside the domain. The main field of application is shape optimization. This paper addresses the case of the steady-state Navier-Stokes equations for an incompressible fluid and a no-slip condition prescribed on the boundary of an arbitrary shaped obstacle. The two and three dimensional cases are treated for several examples of cost functional and a numerical application is presented.
78 citations
TL;DR: A general method for the analysis of finite difference schemes to investigate lattice Boltzmann algorithms for Navier–Stokes problems with Dirichlet boundary conditions is used and the accuracy of the algorithms can be exactly predicted.
Abstract: In this article, we use a general method for the analysis of finite difference schemes to investigate lattice Boltzmann algorithms for Navier–Stokes problems with Dirichlet boundary conditions Several link based boundary conditions for commonly used lattice Boltzmann BGK models are considered With our method, the accuracy of the algorithms can be exactly predicted Moreover, the analytical results can be used to construct new algorithms which is demonstrated with a corrected bounce back rule that requires only local evaluations but still yields second order accuracy for the velocity The analysis is applicable to general geometries and instationary flows
71 citations
TL;DR: In this paper, the biorthogonal polynomials that appear in the theory of coupled random matrices via a Riemann-Hilbert problem are characterized.
65 citations
Book•
28 Dec 2005
TL;DR: The Euler Gamma Function Integral Solutions Expansion in Basis Functions Airy Phase Integral Methods II Bessel Weber-Hermite Whittaker and Watson Inhomogeneous Differential Equations The Riemann Zeta Function Boundary Layer Problems as mentioned in this paper.
Abstract: Dominant Balance Exact Solutions Complex Variables Local Approximate Solutions Phase Integral Methods I Perturbation Theory Asymptotic Evaluation of Integrals The Euler Gamma Function Integral Solutions Expansion in Basis Functions Airy Phase Integral Methods II Bessel Weber-Hermite Whittaker and Watson Inhomogeneous Differential Equations The Riemann Zeta Function Boundary Layer Problems.
59 citations
TL;DR: In this article, the authors investigate the existence of trapped modes in elastic plates of constant thickness, which possess bends of arbitrary curvature and flatten out at infinity; such trapped modes consist of finite energy localized in regions of maximal curvature.
Abstract: We investigate the existence of trapped modes in elastic plates of constant thickness, which possess bends of arbitrary curvature and flatten out at infinity; such trapped modes consist of finite energy localized in regions of maximal curvature. We present both an asymptotic model and numerical evidence to demonstrate the trapping. In the asymptotic analysis we utilize a dimensionless curvature as a small parameter, whereas the numerical model is based on spectral methods and is free of the small-curvature limitation. The two models agree with each other well in the region where both are applicable. Simple existence conditions depending on Poison's ratio are offered, and finally, the effect of energy build-up in a bend when the structure is excited at a resonant frequency is demonstrated.
TL;DR: In this paper, the boundary feedback stabilization of a Timoshenko beam with both ends free was considered, and boundary feedback control law was proposed to make the closed loop system dissipative.
Abstract: In the present paper we consider the boundary feedback stabilization of a Timoshenko beam with both ends free. We propose boundary feedback control law that makes the closed loop system dissipative. Using asymptotic analysis techniques, we give explicit asymptotic formula of eigenvalues of the closed loop system, and prove the Riesz basis property of eigenvectors and generalized eigenvectors. By a detailed analysis of spectrum of the closed loop system, we show that the closed system is exponentially stable.
TL;DR: A large-axial-wavenumber asymptotic analysis of inviscid normal modes in an axisymmetric vortex with a weak axial flow is performed in this paper.
Abstract: A large-axial-wavenumber asymptotic analysis of inviscid normal modes in an axisymmetric vortex with a weak axial flow is performed in this work. Using a WKBJ approach, general conditions for the existence of regular neutral modes are obtained. Dispersion relations are derived for neutral modes confined in the vortex core (‘core modes’) or in a ring (‘ring modes’). Results are applied to a vortex with Gaussian vorticity and axial velocity profiles, and a good agreement with numerical results is observed for almost all values of k. The theory is also extended to deal with singular modes possessing a critical point singularity. We demonstrate that the characteristics for vanishing viscosity of viscous damped normal modes can also be obtained. Known viscous damped eigenfrequencies for the Gaussian vortex without axial flow are, in particular, shown to be predicted well by our estimates. The theory is also shown to provide explanations for a few of their peculiar properties.
TL;DR: This paper is devoted to the asymptotic analysis of a coupled model arising in radiative transfer that consists of a kinetic equation satisfied by the specific intensity of radiation coupled to a diffusion equation satisfying by the material temperature.
Abstract: This paper is devoted to the asymptotic analysis of a coupled model arising in radiative transfer. The model consists of a kinetic equation satisfied by the specific intensity of radiation coupled to a diffusion equation satisfied by the material temperature. The interaction terms take into account both scattering and absorption/emission phenomena, as well as Doppler corrections. Two asymptotic regimes are identified, depending on the scaling assumptions about the physical parameters and observation scales. In the equilibrium regime, the system is driven only by the material temperature which satisfies a nonlinear drift-diffusion equation. In the nonequilibrium regime, the radiation temperature and the material temperature will be coupled by a system of nonlinear drift-diffusion equations.
TL;DR: Two different sets of time scales arising in stiff systems of reaction-diffusion PDEs are examined and it is shown that the two approaches provide a non-stiff simplified system of similar accuracy.
Abstract: Two different sets of time scales arising in stiff systems of reaction-diffusion PDEs are examined; the first due to the reaction term alone and the second due to the interaction of the reaction and diffusion terms. The fastest time scales of each set are responsible for the development of a low dimensional manifold, the characteristics of which depend on the set of time scales considered. The advantages and disadvantages of employing these two manifolds for the simplification of large and stiff systems of reaction-diffusion PDEs are discussed. It is shown that the two approaches provide a non-stiff simplified system of similar accuracy. The approach based on the reaction time scales allows for a simpler construction of the simplified system, while that based on the reaction/diffusion time scales allows for a simpler time marching scheme.
TL;DR: In this paper, the asymptotic behavior of solutions to phase-field equations subject to the Neumann boundary was studied, where a Lojasiewicz-Simon type inequality played an important role.
Abstract: This paper is concerned with the asymptotic behavior of solutions
to the phase-field equations subject to the Neumann boundary
conditions where a Lojasiewicz-Simon type inequality plays an
important role. In this paper, convergence of the solution of this
problem to an equilibrium, as time goes to infinity, is proved.
TL;DR: In this article, the authors derive classical particle, string, and membrane motion equations from a rigorous asymptotic analysis of the Born-Infeld nonlinear electromagnetic theory and show that four rescaled versions of the system has smooth solutions existing in the (finite) time interval where the corresponding limit problems have smooth solutions.
Abstract: We derive classical particle, string, and membrane motion equations from a rigorous asymptotic analysis of the Born–Infeld nonlinear electromagnetic theory. We first add to the Born–Infeld equations the corresponding energy-momentum conservation laws and write the resulting system as a nonconservative symmetric 10×10 system of first-order PDEs. Then we show that four rescaled versions of the system have smooth solutions existing in the (finite) time interval where the corresponding limit problems have smooth solutions. Our analysis is based on a continuation principle previously formulated by Yong for (singular) limit problems.
TL;DR: In this article, the authors study the asymptotic behavior of solutions of the primitive equations (PEs) as the depth of the domain goes to zero and prove that the solution of the PEs can be expanded as a sum of barotropic flow and baroclinic flow up to a uniformly bounded (in time and space) initial time layer.
Abstract: In this article we study the asymptotic behavior of solutions of the primitive equations (PEs) as the depth of the domain goes to zero. We prove that the solutions of the PEs can be expanded as a sum of barotropic flow and baroclinic flow up to a uniformly bounded (in time and space) initial time layer. The barotropic flow is solution of the 2D Navier–Stokes equations with Coriolis force coupled with density. By employing a comparison theorem, the baroclinic flow can be approximated by a quasi-stationary nonlinear GFD-Stokes problem. This article presents a mathematically rigorous justification that the barotropic flow dominates the baroclinic flow in the motion of the atmosphere and ocean.
TL;DR: In this article, a general asymptotic ansatz involving multiple horizontal and vertical as well as multiple time scales is presented. But the main purpose of the general multiple scales ansatz is to provide the means to derive true multiscale models which describe interactions between the various phenomena described by the members of the simplified model family.
Abstract: The first part of this paper reviews the single time scale/multiple length scale low Mach number asymptotic analysis by Klein (1995, 2004). This theory explicitly reveals the interaction of small scale, quasi-incompressible variable density flows with long wave linear acoustic modes through baroclinic vorticity generation and asymptotic accumulation of large scale energy fluxes. The theory is motivated by examples from thermoacoustics and combustion. In an almost obvious way specializations of this theory to a single spatial scale reproduce automatically the zero Mach number variable density flow equations for the small scales, and the linear acoustic equations with spatially varying speed of sound for the large scales. Following the same line of thought we show how a large number of well-known simplified equations of theoretical meteorology can be derived in a unified fashion directly from the three-dimensional compressible flow equations through systematic (low Mach number) asymptotics. Atmospheric flows are. however, characterized by several singular perturbation parameters that appear in addition to the Mach number; and that are defined independently of any particular length or time scale associated with some specific flow phenomenon. These are the ratio of the centripetal acceleration due to the earth's rotation vs. the acceleration of gravity, and the ratio of the sound speed vs. the rotational velocity of points on the equator. To systematically incorporate these parameters in an asymptotic approach, we couple them with the square root of the Mach number in a particular distinguished so that we are left with a single small asymptotic expansion parameter, e. Of course, more familiar parameters, such as the Rossby and Froude numbers may then be expressed in terms of £ as well. Next we consider a very general asymptotic ansatz involving multiple horizontal and vertical as well as multiple time scales. Various restrictions of the general ansatz to only one horizontal, one vertical. and one time scale lead directly to the family of simplified model equations mentioned above. Of course, the main purpose of the general multiple scales ansatz is to provide the means to derive true multiscale models which describe interactions between the various phenomena described by the members of the simplified model family. In this context we will summarize a recent systematic development of multiscale models for the tropics (with Majda).
TL;DR: It is proved that in the case where the two relaxation times converge to zero at the same rate there is a drift correction, in addition to the limiting Ito integral, which is not of Stratonovich type.
Abstract: In this paper we present a rigorous asymptotic analysis for stochastic systems with two fast relaxation times. The mathematical model analyzed in this paper consists of a Langevin equation for the particle motion with time-dependent force constructed through an infinite dimensional Gaussian noise process. We study the limit as the particle relaxation time as well as the correlation time of the noise tend to zero, and we obtain the limiting equations under appropriate assumptions on the Gaussian noise. We show that the limiting equation depends on the relative magnitude of the two fast time scales of the system. In particular, we prove that in the case where the two relaxation times converge to zero at the same rate there is a drift correction, in addition to the limiting Ito integral, which is not of Stratonovich type. If, on the other hand, the colored noise is smooth on the scale of particle relaxation, then the drift correction is the standard Stratonovich correction. If the noise is rough on this scale, then there is no drift correction. Strong (i.e., pathwise) techniques are used for the proof of the convergence theorems.
TL;DR: A novel waterfilling algorithm is provided to determine the data-rate-maximizing transmit signal covariance matrix when only the channel long-term spatial correlations are available at the transmitter.
Abstract: Asymptotic Analysis of MIMO Wireless Systems With Spatial Correlation at the Receiver With a unified approach, this paper investigates the asymptotic performance, or equivalently, the large-system properties, of various point-to-point systems with antenna-array-based multiple-input multiple-output (MIMO) channels having spatial correlations. Using the replica method originally developed for statistical physics, we provide analytical solutions to the input–output mutual information of MIMO systems at the transmit side, with arbitrary inputs, and at the receive side, with either the optimum space–time joint decoding, or various suboptimum spatial equalizers followed by a bank of temporal decoders. Important physical meanings revealed though the analytical solutions to those more practical combinations, such as how the input–output mutual information is affected by the channel spatial correlations, are highlighted along our derivation. Moreover, we provide a novel waterfilling algorithm to determine the data-rate-maximizing transmit signal covariance matrix when only the channel long-term spatial correlations are available at the transmitter.
05 May 2005
TL;DR: It is proved that, under certain condition, optimal smoothing splines converge to some limiting curve as the number of data increases.
Abstract: We consider the problem of designing optimal smoothing spline curves and surfaces for a given set of discrete data. For constructing curves and surfaces, we employ normalized uniform B-splines as the basis functions. First we derive concise expressions for the optimal solutions in the form which can be used easily for numerical computa- tions as well as mathematical analyses. Then, assuming that a set of data in a plane is obtained by sampling some curve with or without noises, we prove that, under certain condition, optimal smoothing splines converge to some limiting curve as the number of data increases. Such a limiting curve is obtained as a functional of given curve to be sampled. The case of surfaces is treated in parallel, and it is shown that the results for the case of curves can be extended to the case of surfaces in a straightforward manner. Keywords: B-splines, optimal smoothing splines, asymptotic analysis, statistical anal- ysis
TL;DR: For linear functional difference equations, this article obtained some results on the asymptotic behavior of solutions, which correspond to a Perron-type theorem for linear ordinary difference equations.
TL;DR: The asymptotic behavior of the solution to the linearized compressible Navier--Stokes equations with highly oscillatory forces in the half-plane with nonslip boundary conditions for small viscosity is studied.
Abstract: We study the asymptotic behavior of the solution to the linearized compressible Navier--Stokes equations with highly oscillatory forces in the half-plane with nonslip boundary conditions for small viscosity The wavelength of oscillation is assumed to be proportional to the square root of the viscosity By means of asymptotic analysis, we deduce that the leading profiles of the solution have four terms: the first one is the outflow satisfying the linearized Euler equations, the second one is an oscillatory wave propagated along the characteristic field tangential to the boundary associated with the linearized Euler operator in the half-plane, the third one is a boundary layer satisfying a linearized Prandtl equation, the fourth one represents the oscillation propagated in the boundary layer, and it is described by an initial-boundary value problem for an Poisson--Prandtl coupled system By using the energy method and mode analysis, we obtain the well-posedness of this Poisson--Prandtl coupled problem, and
TL;DR: The proposed grid coupling is validated by numerical convergence studies indicating, that the coupling does not affect the second-order convergence behavior of the LB algorithm which is observed on uniform grids.
Abstract: A method of coupling grids of different mesh size is developed for classical Lattice-Boltzmann (LB) algorithms on uniform grids. The approach is based on an asymptotic analysis revealing suitable quantities equalized along the grid interfaces for exchanging information between the subgrids. In contrast to other couplings the method works without overlap zones. Moreover the grid velocity (Mach number) is not kept constant, as the time step depends not linearly but quadratically on the grid spacing. To illustrate the basic idea we use a simple LB algorithm solving the advection-diffusion equation. The proposed grid coupling is validated by numerical convergence studies indicating, that the coupling does not affect the second-order convergence behavior of the LB algorithm which is observed on uniform grids. In order to demonstrate its principal applicability to other LB models, the coupling is generalized to the standard D2P9 model for (Navier-)Stokes flow and tested numerically. As we use analytic tools different from the Chapman-Enskog expansion, the theoretical background material is given in two appendices. In particular, the results of numerical experiments are confirmed with a consistency analysis.
TL;DR: An improved version of the Schwarz domain decomposition method is introduced for highly heterogeneous media using new optimized interface conditions specially designed to take into account the heterogeneity between the subdomains on the interfaces.
TL;DR: In this article, the authors explicitly determine the asymptotic behavior of the exponentially small coupling term for generic two-state systems with real-symmetric Hamiltonians, based on a new norm which allows to rigorously implement Darboux' principle.
Abstract: For multi-level time-dependent quantum systems one can construct superadiabatic representations in which the coupling between separated levels is exponentially small in the adiabatic limit. Based on results from [BeTe1] for special Hamiltonians we explicitly determine the asymptotic behavior of the exponentially small coupling term for generic two-state systems with real-symmetric Hamiltonian. The superadiabatic coupling term takes a universal form and depends only on the location and the strength of the complex singularities of the adiabatic coupling function. Our proof is based on a new norm which allows to rigorously implement Darboux' principle, a heuristic guideline widely used in asymptotic analysis.
TL;DR: In this paper, the shape and speed of the solid-melt interface are described at times just before complete freezing takes place, as well as the temperature field in the vicinity of the extinction point.
Abstract: The one-phase Stefan problem for the inward solidification of a three-dimensional body of liquid that is initially at its fusion temperature is considered. In particular, the shape and speed of the solid-melt interface is described at times just before complete freezing takes place, as is the temperature field in the vicinity of the extinction point. This is accomplished for general Stefan numbers by employing the Baiocchi transform. Other previous results for this problem are confirmed, for example the asymptotic analysis reveals the interface ultimately approaches an ellipsoid in shape, and furthermore, the accuracy of these results is improved. The results are arbitrary up to constants of integration that depend physically on both the Stefan number and the shape of the fixed boundary of the liquid region. In general it is not possible to determine this dependence analytically; however, the limiting case of large Stefan number provides an exception. For this limit a rather complete asymptotic picture is presented, and a recipe for the time it takes for complete freezing to occur is derived. The results presented here for fully three-dimensional domains complement and extend those given by McCue et al.[Proc. R. Soc. London A 459 (2003) 977], which are for two dimensions only, and for which a significantly different time dependence occurs.
TL;DR: In this paper, an anisotropic elasticity problem for a body with cavities into which thin rods are inserted, the outer ends of the rods being rigidly fixed, is studied.
Abstract: Asymptotic formulas are obtained for solutions of the anisotropic elasticity problem for a body with cavities into which thin rods are inserted, the outer ends of the rods being rigidly fixed. The surface of the body and the lateral surface of the rods are assumed load-free, but the entire elastic junction is subject to mass forces. The elastic materials are inhomogeneous and the stiffness of the rods may differ greatly from that of the body, their ratio being of the order hγ with an arbitrary exponent γ ∈ ℝ; for γ = 0, the junction is homogeneous. Together with the asymptotic formulas, we construct and justify an asymptotic model of the junction. This model is applicable for a wide range of the exponent γ and preserves the parameter h in the conjugation conditions but is represented by a regularly perturbed problem. Since the leading asymptotic term involves fields with strong singularities, we have to give correct statements of the limit problem for a body with one-dimensional rods. For this purpose, we use the theory of self-adjoint extensions of operators or the technique of weighted spaces with separated asymptotics. The justification of our asymptotic expansions utilizes weighted anisotropic Korn inequalities, which take into account the mutual position of the rods and provide the best possible a priori estimates of the solutions. In contrast to other investigations, we describe, in explicit terms, the dependence of the bounds in the error estimates on the right-hand sides of the original problem. We also discuss the relationship between the asymptotic ansatz formulas and the weighted norms in the asymptotically precise Korn inequality.