Parabolic Layers Research Articles

Numerical analysis of singularly perturbed parabolic reaction diffusion differential difference equations

The authors present numerical analysis of singularly perturbed parabolic problems. The previous papers in this direction focussed on the convection-diffusion problems with regular type of boundary layers. It is very difficult and almost impossible as stated in the literature, e.g, in Chapter 14 of ‘Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions’ (1996) by Miller, John JH and O'Riordan, Eugene and Shishkin, Grigorii I, to capture singularities due to parabolic layers and develop a robust scheme for reaction-diffusion problems with general values of space shifts. The work is in progress now and this is the first paper in this direction. In this work, we are investigating such problems and develop a robust numerical scheme using Mickens's non-standard finite difference scheme and special mesh. Furthermore, numerical examples have been presented to verify the theoretical findings.

Supercloseness of weak Galerkin method for a singularly perturbed convection–diffusion problem in 2D

In this paper we analyze a weak Galerkin method on a Shishkin rectangular mesh for a singularly perturbed convection–diffusion problem in two dimensions, whose solution has one exponential layer and two parabolic layers. The supercloseness of this method is achieved by a specially constructed interpolant. More specifically, the interpolant defined in the interior of each element consists of a vertices-edges-element interpolant inside the layers and a modified Gauß-Radau interpolant outside the layers, while the interpolant defined on the boundary of each element consists of a vertices-edges-element interpolant inside the layers and a weighted L2 projection outside the layers. Moreover, we make use of the over-penalization technique inside the layers, and then prove supercloseness of order k+1/2, even up to almost k+1 under appropriate assumptions. Numerical experiments support the theoretical result.

Uniform convergence of optimal order for a finite element method on a Bakhvalov-type mesh for a singularly perturbed convection-diffusion equation with parabolic layers

Uniform convergence of optimal order for a finite element method on a Bakhvalov-type mesh for a singularly perturbed convection-diffusion equation with parabolic layers

Parameter-Robust Numerical Scheme for Time-Dependent Singularly Perturbed Reaction–Diffusion Problem with Large Delay

ABSTRACTThis work presents the development of numerical scheme for second-order time-dependent singularly perturbed reaction-diffusion problem with large delay in the undifferentiated term. These types of problems arise frequently in many areas of science and engineering that take into consideration the effect of present situation as well as the past history of the physical system. As the characteristics of the reduced problem (𝜀 = 0) corresponding to the original singularly perturbed problem considered here are parallel to the boundary of the domain this implies, parabolic layers exhibit in the solution. In this paper, we initiate the study of parabolic layers together with interior layers in the solution of singularly perturbed parabolic partial differential-difference equations due to propagation of singularity. Proposed numerical scheme comprised of finite difference scheme and piecewise uniform Shishkin mesh. The method is shown to be accurate of order , where M and N are the number of mesh elements in time and spatial direction, respectively. Proposed numerical scheme is proved to be parameter uniform convergent in the maximum norm. Numerical experiments have been performed to show the existence of interior layer due to large state-dependent delay argument in the reaction term and to confirm the predicted theory.

Migration behavior of alkali and alkaline-earth cations in soda-lime silicate glass surface by electrical nanoimprint

One dimensional pattern of 500nm period was formed on a soda-lime silicate glass by the electrical nanoimprint below the glass transition temperature. The imprinted glass surface was analyzed using a transmission electron microscopy and an energy-dispersive X-ray spectroscopy with the cryogenic system. The concentrated parabolic layers of Ca2+ and Mg2+ were formed separately in the Na+ deficient layer. The deficient area both of such alkali and alkaline-earth cations was preferentially removed in an alkali etchant, resulting the enhancement of the aspect ratio of the one-dimensional relief pattern. Therefore, the distribution control of alkaline-earth cations was important for the fabrication of sub-micron scale patterns with high aspect ratio on the glass surface.

A high-resolution Petrov–Galerkin method for the convection–diffusion–reaction problem. Part II—A multidimensional extension

A multidimensional extension of the HRPG method (doi:10.1016/j.cma.2009.10.009) using the lowest order block finite elements is presented. First, we design a nondimensional element number that quantifies the characteristic layers which are found only in higher dimensions. This is done by matching the width of the characteristic layers to the width of the parabolic layers found for a fictitious 1D reaction–diffusion problem. The nondimensional element number is then defined using this fictitious reaction coefficient, the diffusion coefficient and an appropriate element size. Next, we introduce anisotropic element length vectors li and the stabilization parameters αi, βi are calculated along these li. Except for the modification to include the new dimensionless number that quantifies the characteristic layers, the definitions of αi, βi are a direct extension of their counterparts in 1D. Using αi, βi and li, objective characteristic tensors associated with the HRPG method are defined. The numerical artifacts across the characteristic layers are manifested as the Gibbs phenomenon. Hence, we treat them just like the artifacts formed across the parabolic layers in the reaction-dominant case. Several 2D examples are presented that support the design objective—stabilization with high-resolution.

On the method for calculating the reflection function

We consider the problem of calculating the reflection function for the case of radio-wave propagation in a stratified inhomogeneous absorbing ionosphere. A stable method for calculating the reflection function is proposed. Examples of calculations by the proposed method are shown for symmetric and parabolic ionospheric layers.

Layer-adapted meshes and FEM for time-dependent singularly perturbed reaction-diffusion problems

We consider a non-monotone FEM discretisation of a singularly perturbed time-dependent reaction-diffusion problem whose solution exhibits strong parabolic layers. We conduct a stability and convergence analysis for arbitrary meshes. The method is shown to be maximum-norm stable although it is not inverse monotone. The convergence result implies convergence of the method on Shishkin and Bakhvalov meshes uniformly in the perturbation parameter e. Numerical experiments complement our theoretical results.

Interior penalty discontinuous approximations of convection–diffusion problems with parabolic layers

A nonsymmetric discontinuous Galerkin finite element method with interior penalties is considered for two–dimensional convection–diffusion problems with regular and parabolic layers. On an anisotropic Shishkin–type mesh with bilinear elements we prove error estimates (uniformly in the perturbation parameter) in an integral norm associated with this method. On different types of interelement edges we derive the values of discontinuity–penalization parameters. Numerical experiments complement the theoretical results.

Domain decomposition for an advection–diffusion problem with parabolic layers

This paper deals with multidomain domain decomposition algorithms applied to the solution of a singularly perturbed advection–diffusion problem with parabolic layers. Our purpose is to study the algorithms applied to finite difference approximations of the problem. The finite difference approximations make use of special piecewise uniform meshes. We consider two algorithms: the first one is based on decomposition of the computational domain into nonoverlapping vertical subdomains, and second uses decomposition into nonoverlapping horizontal subdomains. Convergence properties of the algorithms are established. Numerical results for a test problem are presented.

A Robust Method of Improved Order for Convection-Diffusion Problems in a Domain with Characteristic Boundaries

We consider a singularly perturbed two-dimensional convection-diffusion problem in a rectangle with flow parallel to the x-axis. On the inflow and outflow boundaries Dirichlet type conditions are imposed and on the characteristic boundaries regular Robin type conditions (including the possibility of Neumann conditions) are given. For small values of the parameter e, regular (strong) and parabolic (weak) layers appear. In this case the parabolic layers do not involve the first term of the asymptotic expansion. For the proposed problem, e-uniformly convergent methods are considered. Note, that for the upwind finite difference scheme the order of e-uniform convergence does not exceed one. Using the High Order Compact (HOC) standard technique, we construct a classical method on piecewise uniform Shishkin meshes having uniform convergence with order 3/2 in the maximum norm. Numerical results are presented and discussed.

Optimal Convergence of Basic Schemes for Elliptic Boundary Value Problems with Strong Parabolic Layers

We consider a convection-diffusion problem with strong parabolic boundary layers and its discretization using upwind finite differences or bilinear finite elements on a layer-adapted mesh. Based on a new decomposition of the solution we are able to prove optimal uniform convergence results.

Directive photonic-bandgap antennas

This paper introduces two new photonic bandgap (PBG) material applications for antennas, in which a photonic parabolic reflector is studied. It is composed of dielectric parabolic layers associated to obtain a PBG material. The frequency gap is used to reflect and focus the electromagnetic waves. This device has been designed using a finite-difference time-domain (FDTD) code. FDTD computations have provided the theoretical reflector's directivity. These results are in good agreement with measurements, and it appears that the PBG reflector presents the same directivity as a metallic parabola. A second application uses a defect PBG material mode associated with a metallic plate to increase the directivity of a patch antenna. We explain the design of such a device and propose experimental results to validate the theoretical analysis.

Necessary conditions for uniform convergence of finite difference schemes for convection-diffusion problems with exponential and parabolic layers

Necessary conditions for uniform convergence of finite difference schemes for convection-diffusion problems with exponential and parabolic layers

Necessary conditions for uniform convergence of finite-difference schems for convection-diffusion problems with exponential and parabolic layers

Necessary conditions for uniform convergence of finite-difference schems for convection-diffusion problems with exponential and parabolic layers

Special Meshes for Finite Difference Approximations to an Advection-Diffusion Equation with Parabolic Layers

In this paper a model problem for fluid flow at high Reynolds number is examined. Parabolic boundary layers are present because part of the boundary of the domain is a characteristic of the reduced differential equation. For such problems it is shown, by numerical example, that upwind finite difference schemes on uniform meshes are not ε-uniformly convergent in the discrete L∞ norm, where ε is the singular perturbation parameter. A discrete L∞ ε-uniformly convergent method is constructed for a singularly perturbed elliptic equation, whose solution contains parabolic boundary layers for small values of the singular perturbation parameter ε. This method makes use of a special piecewise uniform mesh. Numerical results are given that validate the theoretical results, obtained earlier by the last author, for such special mesh methods.

Single-hop F2 propagation above 30 MHz and over distances greater than 4000 km

During recent years of increased solar activity there has been an increase in the number of reports of communications at 50 MHz over distances exceeding 4000 km. This has led to discussion on the propagation modes and to suggestions that the standard textbook explanation fails to explain that such propagation can occur via single-hop F2 modes. It is pointed out that the standard textbook explanation based on parabolic layers predicts such modes. The most relevant ionospheric properties are briefly discussed, and some results obtained using quasi-parabolic segments (QPSs) to represent the ionosphere are presented. The QPS model provides analytical expressions for ray parameters for propagation in spherically stratified ionospheres. It is shown that the lower E and F1 layers can extend the range of F2 propagation beyond 4000 km. >

Structure of the upper teeth of the filefish, Stephanolepis cirrhifer

The teeth of the filefish were investigated with light and scanning electron microscopy. Anterior and posterior rows of pleurodont teeth are present in the maxilla. The teeth in the anterior row possess a large labial and small lingual surface and the teeth in the posterior row possess a small labial and large lingual surface. The enameloid consists of three layers-the outer, middle, and inner layer. The outer layer consists of fine, parallel crystallites oriented perpendicular to the surface. They are similar to the crystallites in enamel derived from ectoderm. In the middle layer the crystallites form parallel bundles that are oriented in the same direction as that of the crystallites in the outer layer. In the inner layer bundles of crystallites interlace with each other. The tubular structures and the terminal branches of the dentinal tubules are located among the crystallites in the inner layer. In the dentine numerous dentinal tubules radiate from the pulp cavity toward the periphery and are seen to curve slightly in an S-shaped course. At the occlusal tip the alternating parabolic layers are seen and these are considered to be mineralized lines. The openings of the dentinal tubules are round or oval, surrounded by interlacing fibrils. The teeth are tightly fixed to the jawbone by bundles of fibrils. At the lower part of the lingual surface of the teeth in the anterior row and the labial surface of the teeth in the posterior row the bundles of fibrils start at the dentine and some fibrils run through connective tissue, while others terminate in projections of the jawbones.(ABSTRACT TRUNCATED AT 250 WORDS)

Seismograms from Epstein Transition Zones

Spectral amplitudes and theoretical seismograms in the vicinity of caustic and critical points are evaluated numerically for acoustic waves propagating in Epstein media. The amplitude decay into the region beyond the caustic is compared with the asymptotic results and good agreement is found at high frequency. At low frequencies the partial reflection enhances the amplitudes at sub-critical angles and the amplitude increases with decreasing thickness of the transition zone. For a thin transition and low frequencies a head wave exists whose amplitude is inversely proportional to frequency. The study of amplitudes near caustics and critical point is of significant theoretical and experimental interest to seismologists. Improved coverage by stations of the USCGS worldwide network has provided better data and their interpretation has lead to more accurate estimates of elastic wave velocities within the Earth (Johnson 1967). In regions of high velocity gradient in the upper mantle caustics are formed and triplications occur on the travel-time curves. In this paper the amplitude properties near caustics and critical point are studied for the Epstein velocity transitions (Epstein 1930). Epstein profiles represent a group of models with transition zones for which solutions of the wave equation exist in terms of familiar tabulated functions. Wave propagation has also been studied analytically for linear transition layers (Nakamura 1964; Gupta 1966; Hirasawa & Berry 1971; Ward 1973) and exponential and parabolic layers (Merzer 1971 ; Rydbeck 1943). Linear layers have discontinuities in velocity gradient which may not be realistic. Merzer’s study is concerned only with head waves and their amplitude dependence on frequency, layer thickness and shape of the transition. Using Epstein models the caustics, head waves and shadow can be studied. Epstein (1930), Rawer (1939) and Phinney (1970) studied the modulus of the reflection coefficient in order to estimate the partial reflection. The modulus of the reflection coefficient itself, however, cannot give a full description of the energy propagation when the waves interact with the strong velocity gradient. That can only be studied when the spectral amplitudes and theoretical seismograms are evaluated. In this paper the exact spectral amplitudes are computed by contour integration using the method of Phinney & Cathles (1969). For evaluation of the theoretical seismograms the technique of Chapman & Phinney (1972) is employed. The amplitude curves are evaluated in the neighbourhood of both ends of the triplication. At high frequencies the results agree with the asymptotic results. At low

Phase path characteristics for oblique ray paths

The paper is concerned with the theory of oblique propagation of high-frequency radio waves via a spherically-stratified ionosphere above a curved earth; the effects of the Earth's magnetic field and of electron collisions are neglected. The variations of phase path length with elevation angle for a fixed frequency, and with frequency for a fixed ground range, are described. The theory is illustrated with results computed for idealised electron-density profiles corresponding to one and two parabolic layers; path integrals are given for the latter model.