In this article, we investigate the time periodic solutions for two-dimensional Navier-Stokes equations with nontrivial time periodic force terms. Under the time periodic assumption of the force term, the existence of time periodic solutions for two-dimensional Navier-Stokes equations has received extensive attention from many authors. With the smallness assumption of the time periodic force, we show that there exists only one time periodic solution and this time periodic solution is globally asymptotically stable in the H1 sense. Without smallness assumption of the force term, there is no stability analysis theory addressed. It is expected that when the amplitude of the force term is increasing, the time periodic solution is no longer asymptotically stable. In the last part of the article, we use numerical experiments to study the bifurcation of the time periodic solutions when the amplitude of the force is increasing. Extrapolating to the heating of the earth by the sun, the bifurcation diagram hints that when the earth receives a relatively small amount of solar energy regularly, the time periodic fluid patterns are asymptotically stable; while/when the earth receives too much solar energy even though in a time periodic way, the time periodic pattern of the fluid motions will lose its stability.
We propose a boundary layer analysis which fits a domain with corners. In particular, we consider nonlinear reaction–diffusion problems posed in a polygonal domain having a small diffusive coefficient ε>0. We present the full analysis of the singular behaviours at any orders with respect to the parameter ε where we use a systematic nonlinear treatment initiated in Jung et al. (2016). The boundary layers are formed near the polygonal boundaries and two adjacent ones overlap at a corner P and the overlapping produces additional layers, the so-called corner layers. It is noteworthy that the boundary layers are also degenerate due to the singularities of the solutions involving a negative power of the radial distance to the corner P which are present in the Laplace operator on a sector (sector corresponding to the part of the polygon near the corner). The corner layers are then designed to absorb both the singularities and the interaction of the two boundary layers at P.
In this article, we consider a singularly perturbed nonlinear reaction-diffusion equation whose solutions display thin boundary layers near the boundary of the domain. We fully analyse the singular behaviours of the solutions at any given order with respect to the small parameter ε, with suitable asymptotic expansions consisting of the outer solutions and of the boundary layer correctors. The systematic treatment of the nonlinear reaction terms at any given order is novel along the singular perturbation analysis. We believe that the analysis can be suitably extended to other nonlinear problems.
We study the asymptotic behavior of the two dimensional Helmholtz scattering problem with high wave numbers in an exterior domain, the exterior of a circle. We impose the Dirichlet boundary condition on the obstacle, which corresponds to an incidental wave. For the outer boundary, we consider the Sommerfeld conditions. Using a polar coordinates expansion, the problem is reduced to a sequence of Bessel equations. Investigating the Bessel equations mode by mode, we find that the solution of the scattering problem converges to its limit solution at a specific rate depending on k.
We consider the Dirichlet boundary value problem for the viscous Burgers’ equation with a time periodic force on a one dimensional finite interval. Under the boundedness assumption on the external force, we prove the existence of the time-periodic solution by using the Galerkin method and Schaefer’s fixed point theorem. Furthermore, we show that this time-periodic solution is unique and time-asymptotically stable in the H1 sense under an additional smallness condition on the external force. It is naturally expected that when the amplitude of the external force increases and crosses a certain critical value, the time-periodic solution is no longer asymptotically stable. In the last part of the article, to support our theory, numerical experiments are carried out to investigate the exchange of stabilities of the time-periodic solutions when the amplitude of the force crosses the first critical value. We numerically find this critical value at which the stable solutions turn into the unstable ones.
The purpose of this paper is to mathematically investigate the formation of a plasma sheath near the surface of a ball-shaped material immersed in a bulk plasma, and to obtain qualitative information of such a plasma sheath layer. Specifically, we study existence and the quasi-neutral limit behavior of the stationary spherical symmetric solutions for the Euler–Poisson equations in a three-dimensional annular domain. We first propose a suitable condition on the velocity at the sheath edge, referred as to Bohm criterion for the annulus, and under this condition together with the constant Dirichlet boundary conditions for the potential, we show that there exists a unique stationary spherical symmetric solution. Moreover, we study the quasi-neutral limit behavior by establishing (Formula presented.) estimate of the difference of the solutions to the Euler–Poisson equations and its quasi-neutral limiting equations, incorporated with the correctors for the boundary layers. The quasi-neutral limit analysis employing the correctors and their pointwise estimates enables us to obtain detailed asymptotic behaviors including the convergence rates in (Formula presented.) and (Formula presented.) norms as well as the thickness of the boundary layers as a consequence of the pointwise estimates.
The article is devoted to prove the existence and regularity of the solutions of the 3D inviscid Linearized Primitive Equations (LPEs) in a channel with lateral periodicity. This was assumed in a previous work (Hamouda et al. in Discret Contin Dyn Syst Ser S 6(2):401–422, 2013) which is concerned with the boundary layers generated by the corresponding viscous problem. Although the equations under investigation here are of hyperbolic type, the standard methods do not apply because of the specificity of the hyperbolic system. A set of non-local boundary conditions for the inviscid LPEs has to be imposed at the lateral boundary of the channel making thus the system well-posed.
In this article, we review recent progresses in boundary layer analysis of some singular perturbation problems. Using the techniques of differential geometry, an asymptotic expansion of reaction-diffusion or heat equations in a domain with curved boundary is constructed and validated in some suitable functional spaces. In addition, we investigate the effect of curvature as well as that of an ill-prepared initial data. Concerning convection-diffusion equations, the asymptotic behavior of their solutions is difficult and delicate to analyze because it largely depends on the characteristics of the corresponding limit problems, which are first order hyperbolic differential equations. Thus, the boundary layer analysis is performed on relatively simpler domains, typically intervals, rectangles, or circles. We consider also the interior transition layers at the turning point characteristics in an interval domain and classical (ordinary), characteristic (parabolic) and corner (elliptic) boundary layers in a rectangular domain using the technique of correctors and the tools of functional analysis. The validity of our asymptotic expansions is also established in suitable spaces
A Compton camera has been suggested for use in single photon emission computed tomography because a conventional gamma camera has low efficiency. Here we consider a cone transform brought about by a Compton camera with line detectors. A cone transform takes a given function on the 3-dimensional space and assigns to it the surface integral of the function over cones determined by the 1-dimensional vertex space, the 1-dimensional central axis, and the 1-dimensional opening angle. We generalize this cone transform to n-dimensional space and provide an inversion formula. Also, numerical simulations are presented to demonstrate our suggested algorithm in three dimensions.
In this article we study the boundary layers for the viscous Linearized Primitive Equations (LPEs) when the viscosity is small. The LPEs are considered here in a cube. Besides the usual boundary layers that we analyze here too, corner layers due to the interaction between the different boundary layers are also studied.
A new semi-analytical time differencing is applied to spectral methods for partial differential equations which involve higher spatial derivatives. This is developed in Jung and Nguyen (J Sci Comput (2015) 63:355–373) based on the classical integrating factor (IF) and exponential time differencing (ETD) methods. The basic idea is approximating analytically the stiffness (fast part) by the so-called correctors (see 1.3 below) and numerically the non-stiffness (slow part) by the IF and ETD, etc. It turns out that rapid decay and rapid oscillatory modes in the spectral methods are well approximated by our corrector methods, which in turn provides better accuracy in the numerical schemes presented in the text. We investigate some nonlinear problems with a quadratic nonlinear term, which makes all Fourier modes interact with each other. We construct the correctors recursively to accurately capture the stiffness in the mode interactions. Polynomial or other types of nonlinear interactions can be tackled in a similar fashion.
We study singularly perturbed time dependent convection-diffusion equations in a circular domain. Considering suitable compatibility conditions, we present convergence results and provide as well approximation schemes and error estimates. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via a specific boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a P1 classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical solution using a quasi-uniform mesh, that is without refinement of the mesh in the boundary layer.
A semi-analytical method is developed based on conventional integrating factor (IF) and exponential time differencing (ETD) schemes for stiff problems. The latter means that there exists a thin layer with a large variation in their solutions. The occurrence of this stiff layer is due to the multiplication of a very small parameter (Formula presented.) with the transient term of the equation. Via singular perturbation analysis, an analytic approximation of the stiff layer, which is called a corrector, is sought for and embedded into the IF and ETD methods. These new schemes are then used to approximate the non-stiff part of the solution. Since the stiff part is resolved analytically by the corrector, the new method outperforms the conventional ones in terms of accuracy. In this paper, we apply our new method for both problems of ordinary differential equations and some partial differential equations.
It has been suggested that a Compton camera should be used in single photon emission computed tomography because a conventional gamma camera has low efficiency. It brings about a cone transform, which maps a function onto the set of its surface integrals over cones determined by the detector position, the central axis, and the opening angle of the Compton camera. We provide inversion formulas using complete Compton data for three- and two-dimensional cases. Numerical simulations are presented to demonstrate the suggested algorithms in dimension two. Also, we discuss other inversions and the stability estimates of a cone transform with a fixed central axis.
Our aim in this article is to study the numerical solutions of singularly perturbed convection–diffusion problems in a circular domain and provide as well approximation schemes, error estimates and numerical simulations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a P1 classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical scheme in a quasi-uniform mesh.
The singularly perturbed problems with a turning point were discussed in . The case where the limit problem is compatible with the given data was fully resolved. However, with limited compatibility conditions on the data, the asymptotic expansions were constructed only up to the order of the level of compatibilities. In this paper, using a smooth cut-off function compactly supported around the turning point we resolve the difficulties incurred from the non-compatible data and finally provide the full asymptotic expansions up to any order.
A hybrid operator splitting method is developed for computations of two-dimensional transverse magnetic Maxwell equations in media with multiple random interfaces. By projecting the solutions into the random space using the polynomial chaos (PC) projection method, the deterministic and random parts of the solutions are solved separately. There are two independent stages in the algorithm: the Yee scheme with domain decomposition implemented on a staggered grid for the deterministic part and the Monte Carlo sampling in the post-processing stage. These two stages of the algorithm are subject of computational studies. A parallel implementation is proposed for which the computational cost grows linearly with the number of random interfaces. Output statistics of Maxwell solutions are obtained including means, variance and time evolution of cumulative distribution functions (CDF). The computational results are presented for several configurations of domains with random interfaces. The novelty of this article lies in using level set functions to characterize the random interfaces and, under reasonable assumptions on the random interfaces (see Figure 1), the dimensionality issue from the PC expansions is resolved (see Sections 1.1.2 and 1.2).
This paper is devoted to boundary layer theory for singularly perturbed convection-diffusion equations in the unit circle. Two characteristic points appear, (±1, 0), in the context of the equations considered here, and singularities may occur at these points depending on the behaviour there of a given function f, namely, the flatness or compatibility of f at these points as explained below. Two previous articles addressed two particular cases:  dealt with the case where the function f is sufficiently flat at the characteristic points, the so-called compatible case;  dealt with a generic non-compatible case (f polynomial). This survey article recalls the essential results from those papers, and continues with the general case (f non-flat and non-polynomial) for which new specific boundary layer functions of parabolic type are introduced in addition.
The goal of this article is to study the boundary layers of reaction-diffusion equations in a circle and provide some numerical applications which utilize the so-called boundary layer elements. Via the boundary layer analysis, we obtain the valid asymptotic expansions at any order and devise boundary layer elements to be conveniently used in the finite element schemes. Using boundary layer elements incorporated in the finite element space, we obtain accurate numerical solutions in a quasi-uniform mesh with convergence of order 2.
In this article, we give an asymptotic expansion, with respect to the viscosity which is considered here to be small, of the solutions of the 3D linearized Primitive Equations (EPs) in a channel with lateral periodicity. A rigorous convergence result, in some physically relevant space, is proven. This allows, among other consequences, to confirm the natural choice of the non- local boundary conditions for the non-viscous PEs.
We study the asymptotic behavior at small diffusivity of the solutions, uε, to a convection-diffusion equation in a rectangular domain. The diffusive equation is supplemented with a Dirichlet boundary condition, which is smooth along the edges and continuous at the corners. To resolve the discrepancy, on ∂, between uε and the corresponding limit solution, u0, we propose asymptotic expansions of uε at any arbitrary, but fixed, order. In order to manage some singular effects near the four corners of , the so-called elliptic and ordinary corner correctors are added in the asymptotic expansions as well as the parabolic and classical boundary layer functions. Then, performing the energy estimates on the difference of uε and the proposed expansions, the validity of our asymptotic expansions is established in suitable Sobolev spaces.
In this article we aim to study finite volume approximations which approximate the solutions of convection-dominated problems possessing the so-called interior transition layers. The stiffness of such problems is due to a small parameter multiplied to the highest order derivative which introduces various transition layers at the boundaries and at the interior points where certain compatibility conditions do not meet. Here, we are interested in resolving interior transition layers at turning points. The proposed semi-analytic method features interior layer correctors which are obtained from singular perturbation analysis near the turning points. We demonstrate this method is efficient, stable and it shows 2nd-order convergence in the approximations
We study the asymptotic behavior, at small viscosity ε, of the Navier-Stokes equations in a 2D curved domain. The Navier-Stokes equations are supplemented with the slip boundary condition, which is a special case of the Navier friction boundary condition where the friction coefficient is equal to two times the curvature on the boundary. We construct an artificial function, which is called a corrector, to balance the discrepancy on the boundary of the Navier-Stokes and Euler vorticities. Then, performing the error analysis on the corrected difference of the Navier-Stokes and Euler vorticity equations, we prove convergence results in the L2 norm in space uniformly in time, and in the norm of H1 in space and L2 in time with rates of order ε3/4 and ε1/4, respectively. In addition, using the smallness of the corrector, we obtain the convergence of the Navier-Stokes solution to the Euler solution in the H1 norm uniformly in time with rate of order ε1/4.
We study the boundary layers and singularities generated by a convection-diffusion equation in a circle with noncompatible data. More precisely, the boundary of the circle has two characteristic points where the boundary conditions and the external data f are not compatible. Very complex singular behaviors are observed, and we analyze them systematically for highly noncompatible data. The problem studied here is a simplified model for problems of major importance in fluid mechanics and thermohydraulics and in physics.
In this article, we consider the barotropic quasigeostrophic equation of the ocean in the context of the β-plane approximation and small viscosity (see, e.g., [21, 22]). The aim is to study the behavior of the solutions when the viscosity goes to zero. To avoid the substantial complications due to the corners (see, e.g., ) which will be addressed elsewhere, we assume periodicity in one direction (0y). The behavior of the solution in the boundary layers at x = 0, 1 necessitate the introduction of several correctors, solving various analogues of the Prandtl equation. Convergence is obtained at all orders even in the nonlinear case. We also establish as an auxiliary result, the C∞ regularity of the solutions of the viscous and inviscid quasigeotrophic equations.
In this article we aim to study the boundary layer generated by a convection diffusion equation in a circle. In the model problem that we consider two characteristic points appear. To the best of our knowledge such boundary layer problems have not been studied in a systematic way yet and we indeed know that very complex situations can occur. In the cases that we consider in the present article certain simplifying compatibility conditions are assumed. Other situations will be studied in forthcoming articles which involve noncompatible data, more general domains or higher order operators.
We study uncertainty bounds and statistics of wave solutions through a random waveguide which possesses certain random inhomogeneities. The waveguide is composed of several homogeneous media with random interfaces. The main focus is on two homogeneous media which are layered randomly and periodically in space. Solutions of stochastic and deterministic problems are compared. The waveguide media parameters pertaining to the latter are the averaged values of the random parameters of the former. We investigate the eigenmodes coupling due to random inhomogeneities in media, i.e. random changes of the media parameters. We present an efficient numerical method via Legendre Polynomial Chaos expansion for obtaining output statistics including mean, variance and probability distribution of the wave solutions. Based on the statistical studies, we present uncertainty bounds and quantify the robustness of the solutions with respect to random changes of interfaces.
Continuing an earlier work in space dimension one, the aim of this article is to present, in space dimension two, a novel method to approximate stiff problems using a combination of (relatively easy) analytical methods and finite volume discretization. The stiffness is caused by a small parameter in the equation which introduces ordinary and corner boundary layers along the boundaries of a two-dimensional rectangle domain. Incorporating in the finite volume space the boundary layer correctors, which are explicitly found by analysis, the boundary layer singularities are absorbed and thus uniform meshes can be preferably used. Using the central difference scheme at the volume interfaces, the proposed scheme finally appears to be an efficient second-order accurate one.
In this work, we present a novel method to approximate stiff problems using a finite volume (FV) discretization. The stiffness is caused by the existence of a small parameter in the equation which introduces a boundary layer. The proposed semi-analytic method consists in adding in the finite volume space the boundary layer corrector which encompasses the singularities of the problem. We verify the stability and convergence of our finite volume schemes which take into account the boundary layer structures. A major feature of the proposed scheme is that it produces an efficient stable second order scheme to be compared with the usual stable upwind schemes of order one or the usual costly second order schemes demanding fine meshes.
We investigate the evolution of the probability distribution function in time for some wave and Maxwell equations in random media for which the parameters, e.g. permeability, permittivity, fluctuate randomly in space; more precisely, two different media interface randomly in space. We numerically compute the probability distribution and density for output solutions. The underlying numerical and statistical techniques are the so-called polynomial chaos Galerkin projection, which has been extensively used for simulating partial differential equations with uncertainties, and the Monte Carlo simulations.
In this article, we discuss reaction-diffusion problems which produce ordinary boundary layers and elliptic corner layers. Using the classical polynomial Q1-finite elements spaces enriched with the so-called boundary layer elements which absorb the singularities due to the boundary and corner layers we are able to attain high numerical accuracies. We essentially obtain ε-uniform approximation errors in a weighted energy norm with significant simplifications in the numerical implementations; here we do not use mesh refinements.
It has been demonstrated that the ordinary boundary layer elements play an essential role in the finite element approximations for singularly perturbed problems producing ordinary boundary layers. Here we revise the element so that it has a small compact support and hence the resulting linear system becomes sparse, more precisely, block tridiagonal. We prove the validity of the revised element for some singularly perturbed convection-diffusion equations via numerical simulations and via the H1 – approximation error analysis. Furthermore due to the compact structure of the boundary layer we are able to prove the L2 – stability analysis of the scheme and derive the L2 – error approximations.
Turning points occur in many circumstances in fluid mechanics. When the viscosity is small, very complex phenomena can occur near turning points, which are not yet well understood. A model problem, corresponding to a linear convection-diffusion equation (e.g., suitable linearization of the Navier-Stokes or B́nard convection equations) is considered. Our analysis shows the diversity and complexity of behaviors and boundary or interior layers which already appear for our equations simpler than the Navier-Stokes or B́nard convection equations. Of course the diversity and complexity of these structures will have to be taken into consideration for the study of the nonlinear problems. In our case, at this stage, the full theoretical (asymptotic) analysis is provided. This study is totally new to the best of our knowledge. Numerical treatment and more complex problems will be considered elsewhere.
In this article, we investigate a way to analyze and approximate singularly perturbed convection-diffusion equations in a channel domain when a nonlinear reaction term with polynomial growth is present. We verify that the boundary layer structures are governed by certain simple recursive linear equations and this simplicity implies explicit pointwise and norm estimates. Furthermore, we can utilize the boundary layer structures (elements) in the finite elements discretizations which lead to the stability in the approximating systems and accurate approximation solutions with an economical mesh design, i.e., uniform mesh.
In this article we discuss singularly perturbed convection-diffusion equations in a channel in cases producing parabolic boundary layers. It has been shown that one can improve the numerical resolution of singularly perturbed problems involving boundary layers, by incorporating the structure of the boundary layers into the finite element spaces, when this structure is available; see e.g. [Cheng, W. and Temam, R. (2002). Comput. Fluid. V.31, 453-466; Jung, C. (2005). Numer. Meth. Partial Differ. Eq. V.21, 623-648]. This approach is developed in this article for a convection-diffusion equation. Using an analytical approach, we first derive an approximate (simplified) form of the parabolic boundary layers (elements) for our problem; we then develop new numerical schemes using these boundary layer elements. The results are performed for the perturbation parameter ε in the range 10-1-10 -15 whereas the discretization mesh is in the range of order 1/10-1/100 in the x-direction and of order 1/10-1/30 in the y-direction. Indications on various extensions of this work are briefly described at the end of the Introduction.
Our aim in this article is to show how one can improve the numerical solution of singularly perturbed problems involving boundary layers. By incorporating the structures of boundary layers into finite element spaces, when this structure is available, we can improve the accuracy of approximate solutions and this results in significant simplifications. In this article we discuss singularly perturbed convection-diffusion equations in a channel producing ordinary boundary layers.
Our aim in this article is to show how one can improve the numerical solution of singularity perturbed problems involving boundary layers. Incorporating the structures of boundary layers into finite element spaces can improve the accuracy of approximate solutions and result in significant simplifications. In this article we discuss convection-diffusion equations in the two-dimensional space with a homogeneous Dirichlet boundary condition and a mixed boundary condition.
In this article, we demonstrate how one can improve the numerical solutions of singularly perturbed problems involving multiple boundary layers by using a combination of analytic and numerical tools. Incorporating the structures of boundary layers into finite element spaces can improve the accuracy of approximate solutions and result in significant simplifications. We discuss here convection-diffusion equations in the case where both ordinary and parabolic boundary layers are present.
In this article, we establish the asymptotic behavior, when the viscosity goes to zero, of the solutions of the Linearized Primitive Equations (LPEs) in space dimension 2. More precisely, we prove that the LPEs solution behaves like the corresponding inviscid problem solution inside the domain plus an explicit corrector function in the neighborhood of some parts of the boundary. Two cases are considered, the subcritical and supercritical modes depending on the fact that the frequency mode is less or greater than the ratio between the reference stratified flow (around which we linearized) and the buoyancy frequency. The problem of boundary layers for the LPEs is of a new type since the corresponding limit problem displays a set of (unusual) nonlocal boundary conditions.
Our aim in this article is to study the interaction of boundary layers and corner singularities in the context of singularly perturbed convection-diffusion equations. For the problems under consideration, we determine a simplified form of the corner singularities and show how to use it for the numerical approximation of such problems in the context of variational approximations using the concept of enriched spaces.