Showing papers on "Method of matched asymptotic expansions published in 2007"
TL;DR: The assumptions and reasoning that yield composite asymptotic expansions for wall turbulence are discussed, with particular attention paid to the scaling quantities that are used to render the variables non-dimensional and of order one.
Abstract: In this article, the assumptions and reasoning that yield composite asymptotic expansions for wall turbulence are discussed. Particular attention is paid to the scaling quantities that are used to render the variables non-dimensional and of order one. An asymptotic expansion is proposed for the streamwise Reynolds stress that accounts for the active and inactive turbulence by using different scalings. The idea is tested with the data from the channel flows and appears to have merit.
55 citations
TL;DR: In this article, matched asymptotic expansions were used to look into the problem of the scattering of plane SH waves by topographic irregularities of a restricted range in an otherwise plane half-space when the characteristic length dimension of the irregularity is much smaller than the wavelength of the incident wave.
Abstract: Summary
The method of matched asymptotic expansions is used to look into the problem of the scattering of plane SH waves by topographic irregularities of a restricted range in an otherwise plane half-space when the characteristic length dimension of the irregularity is much smaller than the wavelength of the incident wave In contrast to previous work the slope of the irregularity remains arbitrary Expressions for the near and far scattered fields are obtained Comparison between this theory and the regular perturbation technique (which also assumes that the irregularity has a small slope) show that both agree when the slope is small but differ in the general case Results are given for irregularities in the shape of triangles, trapezia and semicircles
52 citations
TL;DR: A numerical method is constructed for this problem which involves an appropriate piecewise-uniform mesh on each time subinterval and the difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter.
Abstract: This paper deals with the singularly perturbed initial value problem for a linear first-order delay differential equation. A numerical method is constructed for this problem which involves an appropriate piecewise-uniform mesh on each time subinterval. The difference scheme is shown to converge to the continuous solution uniformly with respect to the perturbation parameter. Numerical results are presented.
47 citations
TL;DR: In this paper, a numerical method based on finite difference scheme and Shishkin mesh for singularly perturbed two second order weakly coupled system of ordinary differential equations with discontinuous source term is presented.
44 citations
Journal Article•
TL;DR: Uniform asymptotic approximations of Green's kernels for boundary value problems of elasticity in singularly perturbed domains containing a small hole are presented.
Abstract: We present uniform asymptotic approximations of Green's kernels for boundary value problems of elasticity in singularly perturbed domains containing a small hole. We consider the cases of two and t ...
42 citations
01 Apr 2007
40 citations
TL;DR: A parameter robust computational method is constructed and it is proved that it gives essentially first order parameter-uniform convergence in the maximum norm.
39 citations
TL;DR: New types of stationary solutions of a one-dimensional driven sixth-order Cahn–Hilliard-type equation that arises as a model for epitaxially growing nanostructures, such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms.
Abstract: New types of stationary solutions of a one-dimensional driven sixth-order Cahn-Hilliard type equation that arises as a model for epitaxially growing nano-structures such as quantum dots, are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. This method yields analytical expressions for far-field behavior as well as the widths of the humps of these spatially non-monotone solutions in the limit of small driving force strength which is the deposition rate in case of epitaxial growth. These solutions extend the family of the monotone kink and antikink solutions. The hump spacing is related to solutions of the Lambert $W$ function. Using phase space analysis for the corresponding fifth-order dynamical system, we use a numerical technique that enables the efficient and accurate tracking of the solution branches, where the asymptotic solutions are used as initial input. Additionally, our approach is first demonstrated for the related but simpler driven fourth-order Cahn-Hilliard equation, also known as the convective Cahn-Hilliard equation.
34 citations
TL;DR: In this article, the dispersion relation corresponding to wave propagation through an array of rigid scatterers of arbitrary shape was obtained for the case that the field equation is the two-dimensional Helmholtz equation.
Abstract: The propagation of waves through a doubly-periodic array of identical rigid scatterers is considered in the case that the field equation is the two-dimensional Helmholtz equation. The method of matched asymptotic expansions is used to obtain the dispersion relation corresponding to wave propagation through an array of scatterers of arbitrary shape that are each small relative to both the wavelength and the array periodicity. The results obtained differ from those obtained from homogenization in that there is no requirement that the wavelength be much smaller than the array periodicity, and hence it is possible to examine phenomena, such as band gaps, that are associated with the array periodicity.
32 citations
TL;DR: In this article, a complete asymptotic expansion for eigenvalues of the Lame system of the linear elasticity in domains with small inclusions in three dimensions was derived by an integral equation formulation of the solutions to the harmonic oscillatory linear elastic equation.
Abstract: We derive a complete asymptotic expansion for eigenvalues of the Lame system of the linear elasticity in domains with small inclusions in three dimensions. By an integral equation formulation of the solutions to the harmonic oscillatory linear elastic equation, we reduce this problem to the study of the characteristic values of integral operators in the complex planes. Generalized Rouche's theorem and other techniques from the theory of meromorphic operator-valued functions are combined with asymptotic analysis of integral kernels to obtain full asymptotic expansions for eigenvalues.
32 citations
TL;DR: In this paper, a matched asymptotic expansion of the Debye length to the cell width is used to derive simplified models of the cell in two distinguished limits and to systematically derive the Butler-Volmer boundary conditions.
Abstract: A (time-dependent) model for an electrochemical cell, comprising a dilute binary electrolytic solution between two flat electrodes, is formulated. The method of matched asymptotic expansions (taking the ratio of the Debye length to the cell width as the small asymptotic parameter) is used to derive simplified models of the cell in two distinguished limits and to systematically derive the Butler–Volmer boundary conditions. The first limit corresponds to a diffusion-limited reaction and the second to a capacitance-limited reaction. Additionally, for sufficiently small current flow/large diffusion, a simplified (lumped-parameter) model is derived which describes the long-time behaviour of the cell as the electrolyte is depleted. The limitations of the dilute model are identified, namely that for sufficiently large half-electrode potentials it predicts unfeasibly large concentrations of the ion species in the immediate vicinity of the electrodes. This motivates the formulation of a second model, for a concentrated electrolyte. Matched asymptotic analyses of this new model are conducted, in distinguished limits corresponding to a diffusion-limited reaction and a capacitance-limited reaction. These lead to simplified models in both of which a system of PDEs, in the outer region (the bulk of the electrolyte), matches to systems of ODEs, in inner regions about the electrodes. Example (steady-state) numerical solutions of the inner equations are presented.
TL;DR: In this article, the sharp interface limit for diffusive interface models with the generalized Navier boundary condition was derived for the moving contact line problem, and the leading order dynamic contact angle is the same as the static one satisfying the Young's equation.
Abstract: Using method of matched asymptotic expansions, we derive the sharp interface limit for the diffusive interface model with the generalized Navier boundary condition recently proposed by Qian, Wang and Sheng in (9, 11) for the moving contact line problem. We show that the leading order problem satisfies a boundary value problem for a coupled Hale-Shaw and Navier-Stokes equations with the interface being a free boundary, and the leading order dynamic contact angle is the same as the static one satisfying the Young's equation.
TL;DR: In this article, the authors present a systematic study of localized asymptotic solutions of the one-dimensional wave equation with variable velocity, which can be used in more complicated cases such as inhomogeneous wave equations, the linear surge problem, the small dispersion case, etc.
Abstract: We present a systematic study of the construction of localized asymptotic solutions of the one-dimensional wave equation with variable velocity. In part I, we discuss the solution of the Cauchy problem with localized initial data and zero right-hand side in detail. Our aim is to give a description of various representations of the solution, their geometric interpretation, computer visualization, and illustration of various general approaches (such as the WKB and Whitham methods) concerning asymptotic expansions. We discuss ideas that can be used in more complicated cases (and will be considered in subsequent parts of this paper) such as inhomogeneous wave equations, the linear surge problem, the small dispersion case, etc. and can eventually be generalized to the 2-(and n-) dimensional cases.
TL;DR: In this article, the free surface deformation and flow field caused by the impulsive horizontal motion of a rigid vertical plate into a horizontal strip of inviscid, incompressible fluid, initially at rest, is studied in the small time limit using the method of matched asymptotic expansions.
Abstract: The free surface deformation and flow field caused by the impulsive horizontal motion of a rigid vertical plate into a horizontal strip of inviscid, incompressible fluid, initially at rest, is studied in the small time limit using the method of matched asymptotic expansions. It is found that three different asymptotic regions are necessary to describe the flow. There is a main, O(1) sized, outer region in which the flow is singular at the point where the free surface meets the plate. This leads to an inner region, centered on the point where the free surface initially meets the plate, with size of O(it log t). To resolve the singularities that arise in this inner region, it is necessary to analyse further the flow in an inner-inner region, with size of O(t), again centered upon the wetting point of the nascent rising jet. The solutions of the boundary value problems in the two largest regions are obtained analytically. The solution of the parameter-free nonlinear boundary value problem that arises in the inner-inner region is obtained numerically.
TL;DR: Equations for the Edgeworth expansion of the distributions of the estimators in exploratory factor analysis and structural equation modeling and results show that asymptotic expansion gives substantial improvement of approximation to the exact distribution constructed by simulations over the usual normal approximation.
Abstract: Equations for the Edgeworth expansion of the distributions of the estimators in exploratory factor analysis and structural equation modeling are given. The equations cover the cases of non-normal data, as well as normal ones with and without known first-order asymptotic standard errors. When the standard errors are unknown, the distributions of the Studentized statistics are expanded. Methods of constructing confidence intervals of population parameters with arbitrary asymptotic confidence coefficients are given using the Cornish-Fisher expansion. Simulations are performed to see the usefulness of the asymptotic expansions in exploratory factor analysis with rotated solutions and confirmatory factor analysis. The results show that asymptotic expansion gives substantial improvement of approximation to the exact distribution constructed by simulations over the usual normal approximation.
TL;DR: The convergence analysis is given and the method is shown to have optimal order convergence; by collocating the perturbed differential equation, which is satisfied by a special cubic spline interpolate of the true solution.
TL;DR: In this article, a closure procedure for the hierarchy of moment equations related to linear systems of ordinary differential equations with a random parametric excitation is introduced, and a generalization of Pringsheim's theorem for continued fractions is used in a proof of the procedure convergence.
TL;DR: In this article, the authors give an asymptotic expansion of the solutions of higher-order Poincare difference equation in terms of the characteristic solutions of the limiting equation.
Abstract: We give an asymptotic expansion of the solutions of higher-order Poincare difference equation in terms of the characteristic solutions of the limiting equation. As a consequence, we obtain an asymptotic description of the solutions approaching a hyperbolic equilibrium of a higher-order nonlinear difference equation with sufficiently smooth nonlinearity. The proof is based on the inversion formula for the z -transform and the residue theorem.
TL;DR: A class of singularly perturbed two point boundary value problems of convection-diffusion type for third-order ordinary differential equations (ODEs) with a small positive parameter multi-plying the highest derivative and a discontinuous source term is considered.
Abstract: A class of singularly perturbed two point boundary value problems (BVPs) of convection-diffusion type for third-order ordinary differential equations (ODEs) with a small positive parameter (e) multi-plying the highest derivative and a discontinuous source term is considered. The BVP is reduced to a weakly coupled system consisting of one first-order ordinary differential equation with a suitable initial condition and one second-order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested. In the proposed method we first find a zero-order asymptotic expansion approximation of the solution of the weakly coupled system. Then the system is decoupled by replacing the first component of the solution by its zero-order asymptotic expansion approximation in the second equation. Then the second equation is solved by a finite difference method on a Shishkin mesh (a fitted mesh method). Examples are provided to illustrate the method.
TL;DR: In this paper, the evolution of long nonlinear Rossby waves in a sheared zonal current in the regime where a competition sets in between weak nonlinearity and weak dispersion is considered.
Abstract: This study considers the evolution of long nonlinear Rossby waves in a sheared zonal current in the regime where a competition sets in between weak nonlinearity and weak dispersion. The traditional assumption of a weak amplitude breaks down when the wave speed equals the mean-flow velocity at a certain latitude, due to the appearance of a singularity in the leading order equation, which strongly modifies the flow in the critical layer. Here, nonlinear effects are invoked to resolve this singularity, since the relevant geophysical flows have high Reynolds numbers. Viscosity is introduced in order to render the nonlinear critical-layer solution unique, but the inviscid limit is eventually taken. By the method of matched asymptotic expansions, this inner flow is matched at the edges of the critical layer with the outer flow. This theory is relevant for the phenomenon of Rossby wave breaking and eventual saturation into a nonlinear wave. Spatially localized solutions are described by a Korteweg-de Vries equation, modified by new nonlinear terms, depending on the critical-layer shape. These lead to depression or elevation solitary waves.
Journal Article•
TL;DR: After developing the tools required for application of the fixed point theory in the investigation, some general results about the long-time behavior of solutions of n-th order nonlinear differential equations are presented.
Abstract: We discuss a number of issues important for the asymptotic integration of ordinary differential equations. After developing the tools required for application of the fixed point theory in the investigation, we present some general results about the long-time behavior of solutions of n-th order nonlinear differential equations with an emphasis on the existence of polynomial-like solutions, the asymptotic representation for the derivatives and the effect of perturbations upon the asymptotic behavior of solutions.
TL;DR: In this paper a computational method named as ''An asymptotic finite element method'' for solving singularly perturbed Boundary Value Problems (BVPs) with discontinuous source term and a small positive parameter multiplying the highest derivative is presented.
14 Mar 2007
TL;DR: In this article, the authors considered the Cauchy problem with a singularly perturbed integro-differential equation with a nonlinear integral Fredholm operator, and constructed the asymptotics of the solution to the problem by the method of boundary functions.
Abstract: An initial problem is considered for an ordinary singularly perturbed integro-differential equation with a nonlinear integral Fredholm operator. The case when the reduced equation has a smooth solution is investigated, and the solution to the reduced equation with a corner point is analyzed. The asymptotics of the solution to the Cauchy problem is constructed by the method of boundary functions. The asymptotics is validated by the asymptotic method of differential inequalities developed for a new class of problems.
TL;DR: In this article, the authors design and implement two non-standard finite difference methods (NSFDMs) to solve singularly perturbed advection-reaction equations (SPARE).
Abstract: We design and implement two non-standard finite difference methods (NSFDMs) to solve singularly perturbed advection–reaction equations (SPARE). Our methods constitute a big plus to the class of those ‘rare’ fitted operator methods, which can be extended to singularly perturbed partial differential equations. Unlike the standard finite difference methods (SFDMs), the NSFDMs designed in this paper allow the time and the space step sizes to vary independently of one another and of the parameter e in the SPARE under consideration. The NSFDMs replicate the linear stability properties of the fixed points of the continuous problem. Furthermore, these methods preserve the positivity and boundedness properties of the exact solution. Numerical simulations that confirm the theoretical results are presented. Copyright © 2007 John Wiley & Sons, Ltd.
Book•
20 Aug 2007
Abstract: Lattice-Boltzmann algorithms represent a quite novel class of numerical schemes, which are used to solve evolutionary partial differential equations (PDE). In contrast to finite difference and finite element schemes, lattice-Boltzmann methods rely on a mesoscopic (kinetic) approach. The essential idea consists in setting up an artificial, grid-based particle dynamics, which is chosen such that appropriate averages provide approximate solutions of a certain PDE, typically in the area of fluid dynamics. As lattice-Boltzmann schemes are closely related to finite velocity Boltzmann equations being singularly perturbed by special scalings, their consistency is not obvious, however. This work is concerned with the analysis of lattice-Boltzmann methods, where a particular interest lies in some numeric phenomena like initial layers, multiple time scales and boundary layers. As major analytic tool, regular expansions (Hilbert expansion) are employed to establish consistency. Exemplarily, two and three population algorithms are studied in one space dimension, mostly discretizing the advection-diffusion equation. It is shown how these ‘model schemes’ can be derived from two dimensional schemes in the case of special symmetries. The analysis of the schemes is preceded by an examination of the singular limit being characteristic of the corresponding scaled finite velocity Boltzmann equations. Convergence proofs are obtained using a Fourier series approach and alternatively a general regular expansion combined with an energy estimate. The appearance of initial layers is investigated by multiscale and irregular expansions. Among others, a hierarchy of equations is found which gives insight into the internal coupling of the initial layer and the regular part of the solution. Next, the consistency of the model algorithms is considered followed by a discussion of stability. Apart from proving stability for several cases entailing convergence as byproduct, the spectrum of the evolution operator is examined in detail. Based on this, it is shown that the CFL-condition is necessary and sufficient for stability in the case of a two population algorithm discretizing the advection equation. Furthermore, the presentation touches upon the question whether reliable stability statements can be obtained by rather formal arguments. To gather experience and prepare future work, numeric boundary layers are analyzed in the context of a finite difference discretization for the one-dimensional Poisson equation.
TL;DR: In this article, the Dirichlet problem for a system of two singularly perturbed elliptic reaction-diffusion equations is considered, where the higher order derivatives of the ith equation are multiplied by the perturbation parameter ǫi2 (i = 1, 2).
Abstract: In a rectangle, the Dirichlet problem for a system of two singularly perturbed elliptic reaction-diffusion equations is considered. The higher order derivatives of the ith equation are multiplied by the perturbation parameter ɛi2 (i = 1, 2). The parameters ɛi take arbitrary values in the half-open interval (0, 1]. When the vector parameter ɛ = (ɛ1, ɛ2) vanishes, the system of elliptic equations degenerates into a system of algebraic equations. When the components ɛ1 and (or) ɛ2 tend to zero, a double boundary layer with the characteristic width ɛ1 and ɛ2 appears in the vicinity of the boundary. Using the grid refinement technique and the classical finite difference approximations of the boundary value problem, special difference schemes that converge ɛ-uniformly at the rate of O(N−2ln2N) are constructed, where N = min Ns, Ns + 1 is the number of mesh points on the axis xs.
TL;DR: In this paper, the exponential stability of singularly perturbed impulsive delay differential equations (SPIDDEs) is studied and a sufficient condition is provided to ensure that any solution of SPIDDE is exponentially stable for sufficiently small e > 0.
TL;DR: In this paper, a bound on the Gevrey type of the Taylor development of canard manifolds at degenerate planar turning points is given. But this bound depends on the phase space at the turning point.