Showing papers on "Method of matched asymptotic expansions published in 2009"

Solving pdes in complex geometries: a diffuse domain approach.

Abstract: We extend previous work and present a general approach for solving partial differential equations in complex, stationary, or moving geometries with Dirichlet, Neumann, and Robin boundary conditions. Using an implicit representation of the geometry through an auxilliary phase field function, which replaces the sharp boundary of the domain with a diffuse layer (e.g. diffuse domain), the equation is reformulated on a larger regular domain. The resulting partial differential equation is of the same order as the original equation, with additional lower order terms to approximate the boundary conditions. The reformulated equation can be solved by standard numerical techniques. We use the method of matched asymptotic expansions to show that solutions of the re-formulated equations converge to those of the original equations. We provide numerical simulations which confirm this analysis. We also present applications of the method to growing domains and complex three-dimensional structures and we discuss applications to cell biology and heteroepitaxy.

A new phase-field model for strongly anisotropic systems

Abstract: We present a new phase-field model for strongly anisotropic crystal and epitaxial growth using regularized, anisotropic Cahn–Hilliard-type equations. Such problems arise during the growth and coarsening of thin films. When the anisotropic surface energy is sufficiently strong, sharp corners form and unregularized anisotropic Cahn–Hilliard equations become ill-posed. Our models contain a high-order Willmore regularization, where the square of the mean curvature is added to the energy, to remove the ill-posedness. The regularized equations are sixth order in space. A key feature of our approach is the development of a new formulation in which the interface thickness is independent of crystallographic orientation. Using the method of matched asymptotic expansions, we show the convergence of our phase-field model to the general sharp-interface model. We present two- and three-dimensional numerical results using an adaptive, nonlinear multigrid finite-difference method. We find excellent agreement between the dynamics of the new phase-field model and the sharp-interface model. The computed equilibrium shapes using the new model also match a recently developed analytical sharp-interface theory that describes the rounding of the sharp corners by the Willmore regularization.

Approximate Solutions for Pressure Buildup During CO2 Injection in Brine Aquifers

Abstract: If geo-sequestration of CO 2 is to be employed as a key emissions reduction method in the global effort to mitigate against climate change, simple yet robust screening of the risks of disposal in brine aquifers will be needed. There has been significant development of simple analytical and semi-analytical techniques to support screening analysis and performance assessment for potential carbon sequestration sites. These techniques have generally been used to estimate the size of CO 2 plumes for the purpose of leakage rate estimation. A common assumption is that both the fluids and the geological formation are incompressible. Consequently, calculation of pressure distribution requires the specification of an arbitrary radius of influence. In this article, a new similarity solution is derived using the method of matched asymptotic expansions. A large time approximation of this solution is then extended to account for inertial effects using the Forchheimer equation. By allowing for slight compressibility in the fluids and formation, the solutions improve on previous work by not requiring the specification of an arbitrary radius of influence. The validity of both solutions is explored by comparison with equivalent finite difference solutions, revealing that the new method can provide robust and mathematically rigorous solutions for screening level analysis, where numerical simulations may not be justified or cost effective.

Diffusion on a Sphere with Localized Traps : Mean First Passage Time, Eigenvalue Asymptotics, and Fekete Points

Abstract: A common scenario in cellular signal transduction is that a diffusing surface-bound molecule must arrive at a localized signaling region on the cell membrane before the signaling cascade can be completed. The question then arises of how quickly such signaling molecules can arrive at newly formed signaling regions. Here, we attack this problem by calculating asymptotic results for the mean first passage time for a diffusing particle confined to the surface of a sphere, in the presence of N partially absorbing traps of small radii. The rate at which the small diffusing molecule becomes captured by one of the traps is determined by asymptotically calculating the principal eigenvalue for the Laplace operator on the sphere with small localized traps. The asymptotic analysis relies on the method of matched asymptotic expansions, together with detailed properties of the Green's function for the Laplacian and the Helmholtz operators on the surface of the unit sphere. The asymptotic results compare favorably with ...

On Step-Like Contrast Structure of Singularly Perturbed Systems

Abstract: The existence of a step-like contrast structure for a class of high-dimensional singularly perturbed system is shown by a smooth connection method based on the existence of a first integral for an associated system. In the framework of this paper, we not only give the conditions under which there exists an internal transition layer but also determine where an internal transition time is. Meanwhile, the uniformly valid asymptotic expansion of a solution with a step-like contrast structure is presented.

The initial stage of dam-break flow

Abstract: The liquid flow and the free surface shape during the initial stage of dam breaking are investigated. The method of matched asymptotic expansions is used to derive the leading-order uniform solution of the classical dam-break problem. The asymptotic analysis is performed with respect to a small parameter which characterizes the short duration of the stage under consideration. The second-order outer solution is obtained in the main flow region. This solution is not valid in a small vicinity of the intersection point between the initially vertical free surface and the horizontal rigid bottom. The dimension of this vicinity is estimated with the help of a local analysis of the outer solution close to the intersection point. Stretched local coordinates are used in this vicinity to resolve the flow singularity and to derive the leading-order inner solution, which describes the formation of the jet flow along the bottom. It is shown that the inner solution is self-similar and the corresponding boundary-value problem can be reduced to the well-known Cauchy–Poisson problem for water waves generated by a given pressure distribution along the free surface. An analysis of the inner solution reveals the complex shape of the jet head, which would be difficult to simulate numerically. The asymptotic solution obtained is expected to be helpful in the analysis of developed gravity-driven flows.

Multiple asymptotic solutions for axially travelling waves in porous channels

Abstract: Travelling waves in confined enclosures, such as porous channels, develop boundary layers that evolve over varying spatial scales. The present analysis employs a technique that circumvents guessing of the inner coordinate transformations at the forefront of a multiple-scales expansion. The work extends a former study in which a two-dimensional oscillatory solution was derived for the rotational travelling wave in a porous channel. This asymptotic solution was based on a free coordinate that could be evaluated using Prandtl's principle of matching with supplementary expansions. Its derivation required matching the dominant term in the multiple-scales expansion to an available Wentzel-Kramers-Brillouin (WKB) solution. Presently, the principle of least singular behaviour is used. This approach leads to a multiple-scales approximation that can be obtained independently of supplementary expansions. Furthermore, a procedure that yields different types of WKB solutions is described and extended to arbitrary order in the viscous perturbation parameter. Among those, the WKB expansion of type I is shown to exhibit an alternating singularity at odd orders in the perturbation parameter. This singularity is identified and suppressed using matched asymptotic tools. In contrast, the WKB expansion of type II is found to be uniformly valid at any order. Additionally, matched asymptotic, WKB and multiple-scales expansions are developed for several test cases. These enable us to characterize the essential vortico-acoustic features of the axially travelling waves in a porous channel. All solutions are numerically verified, compared and discussed.

Asymptotic behavior of solutions to drift-diffusion system with generalized dissipation

Abstract: We show the global existence and asymptotic behavior of solutions for the Cauchy problem of a nonlinear parabolic and elliptic system arising from semiconductor model. Our system has generalized dissipation given by a fractional order of the Laplacian. It is shown that the time global existence and decay of the solutions to the equation with large initial data. We also show the asymptotic expansion of the solution up to the second terms as t → ∞.

On the multiple scales perturbation method for difference equations

Abstract: In the classical multiple scales perturbation method for ordinary difference equations (O Delta Es) as developed in 1977 by Hoppensteadt and Miranker, difference equations (describing the slow dynamics of the problem) are replaced at a certain moment in the perturbation procedure by ordinary differential equations (ODEs). Taking into account the possibly different behavior of the solutions of an O Delta E and of the solutions of a nearby ODE, one cannot always be sure that the constructed approximations by the Hoppensteadt-Miranker method indeed reflect the behavior of the exact solutions of the O Delta Es. For that reason, a version of the multiple scales perturbation method for O Delta Es will be presented and formulated in this paper completely in terms of difference equations. The goal of this paper is not only to present this method, but also to show how this method can be applied to regularly perturbed O Delta Es and to singularly perturbed, linear O Delta Es.

Initial value technique for singularly perturbed two point boundary value problems using an exponentially fitted finite difference scheme

Abstract: In this paper, we present an approximate method (Initial value technique) for the numerical solution of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations having a boundary layer at one end (left or right) point. It is motivated by the asymptotic behavior of singular perturbation problems. The original problem is reduced to an asymptotically equivalent first order initial value problem by approximating the zeroth order term by outer solution obtained by asymptotic expansion, and then this initial value problem is solved by an exponentially fitted finite difference scheme. Some numerical examples are given to illustrate the given method. It is observed that the presented method approximates the exact solution very well for crude mesh size h.

Asymptotic expansion of semi-Markov random evolutions

Abstract: Regular and singular parts of asymptotic expansions of semi-Markov random evolutions are given. Regularity of boundary conditions is shown. An algorithm for calculation of initial conditions is proposed.

An Exponentially Fitted Method for Singularly Perturbed Delay Differential Equations

Abstract: This paper deals with singularly perturbed initial value problem for linear first-order delay differential equation. An exponentially fitted difference scheme is constructed in an equidistant mesh, which gives first-order uniform convergence in the discrete maximum norm. The difference scheme is shown to be uniformly convergent to the continuous solution with respect to the perturbation parameter. A numerical example is solved using the presented method and compared the computed result with exact solution of the problem.

Construction of uniform asymptotic solutions of wave-type differential equations by methods of catastrophe theory

Abstract: A mathematical apparatus for solving differential equations of special type by the methods of main, edge, and corner catastrophes is developed The fundamentals of the wave catastrophe theory are considered, including the classification and methods of constructing uniform asymptotics used to describe the structure of wave fields in these domains, together with an analysis of the structure of the field Classes of special functions used to construct uniform asymptotic expansions of wave fields are generally described together with the properties of these classes and the methods of computation

B-spline solution of a singularly perturbed boundary value problem arising in biology

Abstract: We use B-spline functions to develop a numerical method for solving a singularly perturbed boundary value problem associated with biology science We use B-spline collocation method, which leads to a tridiagonal linear system The accuracy of the proposed method is demonstrated by test problems The numerical result is found in good agreement with exact solution

Ważewski’s method for systems of dynamic equations on time scales

Abstract: In the paper the Wazewski’s method, which is well-known for ordinary differential equations, is developed for a system of dynamic equations on an arbitrary time scale. Sufficient conditions guaranteeing the existence of at least one solution with a graph staying in a previously defined open set are derived. This result, generalizing some previous results concerning the asymptotic behavior of solutions of discrete equations, is suitable for investigating of asymptotic behavior of solutions of dynamical systems. A relevant example is considered.

Comprehensive model of electrokinetic flow and migration in microchannels with conductivity gradients

Abstract: A comprehensive model of electrokinetic flow and transport of electrolytes in microchannels with conductivity gradients is developed. The electrical potential is modeled by a combination of an electrostatic and an electrodynamic approach. The fluid dynamics are described by the Navier–Stokes equations, extended by an electrical force term. The chemistry of the system is represented by source terms in the mass transport equations, derived from an equilibrium approach. Moreover, the interaction between ionic species concentration and physicochemical properties of the microchannel substrate (i.e. zeta-potential) is taken into consideration by an empirical approach. Approximate analytical solutions for all variables are found which are valid within the electrical double layer. By using the method of matched asymptotic expansions, these solutions provide boundary conditions for the numerical simulation of the bulk liquid. The models are implemented in a Finite-Element-Code. As an example, simulations of an electrophoretic injection/separation process in a micro-electrophoresis device are performed. The results of the simulations show the strong coupling between the involved physicochemical phenomena. Simulations with a constant and a concentration-depend zeta-potential clarify the importance of a proper modeling of the physicochemical substrate characteristics.

Numerical treatment of singularly perturbed two point boundary value problems using initial-value method

Abstract: In this paper, we describe an initial-value method for linear and nonlinear singularly perturbed boundary value problems in the interval [p,q]. For linear problems, the required approximate solution is obtained by solving the reduced problem and one initial-value problems directly deduced from the given problem. For nonlinear problems the original second-order nonlinear problem is linearized by using quasilinearization method. Then this linear problem is solved as previous method. The present method has been implemented on several linear and non-linear examples which approximate the exact solution. We also present the approximate and exact solutions graphically.

Asymptotic behavior of solutions to elliptic equations in a coated body

Abstract: We consider the Dirichlet boundary-value problem for a class of elliptic equations in a domain surrounded by a thin coating with the thickness $\delta$ and the thermal conductivity $\sigma$. By virtue of a new method we further investigate the results of Brezis, Caffarelli and Friedman [3] in three respects. If the integral of the source term on the interior domain is zero, we study the asymptotic behavior of the solution in the case of $\delta^2$»$\sigma$, $\delta^2$~$\sigma$ and $\delta^2$«$\sigma$ as $\delta$ and $\sigma$ tend to zero, respectively. Also we derive the optimal blow-up rate that was not given in [3]. Finally, in the case of the so-called "optimally aligned coating", i.e., if the thermal tensor matrix of the coating is spatially varying and its smallest eigenvalue has an eigenvector normal to the body at all boundary points, we obtain the asymptotic behavior of the solution by assuming only the smallest eigenvalue is of the same order as $\sigma$.

Examples Illustrating the Use of Renormalization Techniques for Singularly Perturbed Differential Equations

Abstract: With nine examples, we seek to illustrate the utility of the Renormalization Group approach as a unification of other asymptotic and perturbation methods.

Inertial migration of sedimenting particles in a suspension flow through a Hele-Shaw cell

Abstract: Within the framework of the model of two interpenetrating continua, a horizontal laminar dilute-suspension flow in a vertical Hele-Shaw cell is investigated. Using the method of matched asymptotic expansions, an asymptotic model of the transverse migration of sedimenting particles is constructed. The particle migration in the horizontal section of the cell is caused by an inertial lateral force induced by the particle sedimentation and the shear flow of the carrier phase. A characteristic longitudinal length scale is determined, on which the particles migrate across the slot through a distance of the order of the slot half-width. The evolution of the particle number concentration and velocity fields along the channel is studied using the full Lagrangian method. Depending on the particle inertia parameter, different particle migration regimes (with and without crossing of the channel central plane by the particles) are detected. A critical value of the particle inertia parameter corresponding to the change in migration regime is found analytically. The possibility of intersection of the particle trajectories and the formation of singularities in the particle number concentration is demonstrated.

Interplay between degenerate convergence of semigroups and asymptotic analysis: a study of a singularly perturbed abstract telegraph system

Abstract: Singularly perturbed evolution equations may be analyzed by various techniques. In this paper, we focus on ideas of asymptotic analysis and those akin to the Trotter–Kato approximation theorems. Using as an example an abstract telegraph-type system, we show how these two approaches intertwine and complement each other. The paper also may be seen as a continuation of Kato’s project (in Turbululence and Navier–Stokes Equations Theory. Proc. Conf. Univ. Paris-Sud, Orsay 1975, Lect. Notes Math., Springer, Berlin, vol 565, pp. 104–112, 1976) taking account of recent results.

Boundary integral equations for two‐dimensional low Reynolds number flow past a porous body

Abstract: In this paper we use the method of matched asymptotic expansions in order to study the two-dimensional steady flow of a viscous incompressible fluid at low Reynolds number past a porous body of arbitrary shape. One assumes that the flow inside the porous body is described by the Brinkman model, i.e. by the continuity and Brinkman equations, and that the velocity and boundary traction fields are continuous across the interface between the fluid and porous media. By considering some indirect boundary integral representations, the inner problems are reduced to uniquely solvable systems of Fredholm integral equations of the second kind in some Sobolev or Holder spaces, while the outer problems are solved by using the singularity method. It is shown that the force exerted by the exterior flow on the porous body admits an asymptotic expansion with respect to low Reynolds number, whose terms depend on the solutions of the abovementioned system of boundary integral equations. In addition, the case of small permeability of the porous body is also treated. Copyright © 2008 John Wiley & Sons, Ltd.

The inertial lift on a spherical particle settling in a horizontal viscous flow through a vertical slot

Abstract: The inertial lift force exerted on a small rigid sphere settling due to gravity in a horizontal channel flow between vertical walls is investigated. The method of matched asymptotic expansions is used to obtain solutions for the disturbance flow on the length scales of the particle radius and the channel width (inner and outer regions, respectively). The channel Reynolds number is finite, while the particle Reynolds numbers that are based on the slip velocity and the mean shear rate are small. The inner flow is described by the linear Stokes equations. The outer problem is governed by the linear Oseen-like equations with the particle effect approximated by a point force. The outer equations are solved numerically using the two-dimensional Fourier transform of the disturbance velocity field. The lift coefficient is evaluated as a function of governing dimensionless parameters: the particle coordinate across the channel, the channel Reynolds number, and the slip parameter. The particle always migrates away from the walls, with an equilibrium position being on the channel centerline. Close to the walls, the lift coefficient is the same regardless of the slip velocity and the channel Reynolds number. At large channel Reynolds numbers, a local maximum of the migration velocity forms near the channel centerline due to a combined effect of the slip, the linear shear, and the curvature of the undisturbed velocity profile. The results obtained are extended to the case when the drag on a particle has components both parallel and perpendicular to the undisturbed flow. One of primary applications of the results is modeling of the cross-flow migration of settling particles during particle transport in a hydraulic fracture.

Self-consistent poloidal electric field and neoclassical angular momentum flux

Abstract: A complete expression is obtained for the poloidal variation of the electrostatic potential in the banana regime for large aspect ratio flux surfaces using the method of matched asymptotic expansions. The result exhibits a finite discontinuity at the innermost point of a flux surface instead of a divergence as previously reported. Using this expression in combination with the solution of the linearized drift kinetic equation with a model collision operator, the part of the toroidal angular momentum flux due to the poloidal electric field is calculated. The result is larger than the one in existing works, which neglect the poloidal electric field, by the order of the square root of the aspect ratio.

Rational-Harmonic Balancing Approach to Nonlinear Phenomena Governed by Pendulum-Like Differential Equations

Abstract: This paper presents a new approach for solving accurate approximate analytical solutions for nonlinear phenomena governed by pendulum-like differential equations. The new approach couples Taylor series expansion with rational harmonic balancing. An approximate rational solution depending on a small parameter is considered. After substituting the approximate solution into the governing differential equation, this equation is expanded in Taylor series of the parameter prior to harmonic balancing. The approach gives a cubic equation, which must be solved in order to obtain the value of the small parameter. A method for transforming this cubic equation into a linear equation is presented and discussed. Using this approach, accurate approximate analytical expressions for period and periodic solutions are obtained. We also compared the Fourier series expansions of the analytical approximate solution and the exact one. This allowed us to compare the coefficients for the different harmonic terms in these solutions. These analytical approximations may be of interest for those researchers working in nonlinear physical phenomena governed by pendulum-like differential equations in fields such as classical mechanics, vibrations, acoustics, electromagnetism, electronics, superconductivity, optics, gravitation, and others.