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In this paper we deal with solving robustly and efficiently one-dimensional linear parabolic singularly perturbed systems of convection-diffusion type, where the diffusion parameters can be different at each equation and even they can have different orders of magnitude. The numerical algorithm combines the classical upwind finite difference scheme to discretize in space and the fractional implicit Euler method together with an appropriate splitting by components to discretize in time. We prove that if the spatial discretization is defined on an adequate piecewise uniform Shishkin mesh, the fully discrete scheme is uniformly convergent of first order in time and of almost first order in space. The technique used to discretize in time produces only tridiagonal linear systems to be solved at each time level; thus, from the computational cost point of view, the method we propose is more efficient than other numerical algorithms which have been used for these problems. Numerical results for several test problems are shown, which corroborate in practice both the uniform convergence and the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

In this work, we are interested in approximating the solution of 2D parabolic singularly perturbed problems of convection–diffusion type. The convective term of the differential equation, associated to the initial and boundary value problem, is such that each one of its components has an interior simple turning point, which can be of attractive or repulsive type. We describe a numerical method to discretize the continuous problem, which combines the fractional implicit Euler method, defined on a uniform mesh, to discretize in time, and the classical upwind finite difference scheme, defined on a nonuniform mesh of Shishkin type, to discretize in space. The fully discrete algorithm has a low computational cost. From a numerical point of view, we see that the method is efficient and uniformly convergent with respect to the diffusion parameter in both cases when the source term is a continuous function or it has a first kind discontinuity at the turning points. Some numerical results for different test problems are showed; from them, we deduce the good properties of the numerical method. [ABSTRACT FROM AUTHOR]

In this paper we analyze the uniform convergence of a numerical method designed to approximate efficiently the solution of 2D parabolic singularly perturbed problems of reaction diffusion type. The method combines a modified fractional implicit Euler method to discretize in time, and the classical central finite difference scheme, on a special nonuniform mesh, to discretize in space. The resulting fully discrete scheme is uniformly convergent with respect to the diffusion parameter. The analysis of the convergence is made by using a two step technique, which discretizes first in time and later on in space. We show the order reduction phenomenon associated to the fractional implicit Euler method, which typically appears if the boundary conditions are time dependent and a natural evaluation of them is done. An appropriate choice for the boundary conditions is proposed and analyzed in detail, proving that the order reduction can be removed. Some numerical tests show the practical effects of our method; as well, we compare it with the classical choice for the boundary data in terms of the uniform consistency and the order of uniform convergence of the numerical scheme. [ABSTRACT FROM AUTHOR]

In this work we analyze the uniform convergence of some monotone finite difference schemes which are used to semidiscretize in space time dependent two dimensional singularly perturbed parabolic problems of convection–diffusion or reaction–diffusion type. The analysis technique combines a suitable semidiscrete maximum principle joint to some techniques used in the study of the convergence of well known numerical schemes developed to solve elliptic singularly perturbed problems. We focus our attention to standard central or upwind finite difference schemes defined on special nonuniform meshes of Shishkin type. Nevertheless, the technique can be easily extended to other classical discretization methods like, for example, fitted operator methods. We prove that the stiff initial value problems resulting of the spatial semidiscretization processes, have a unique solution which converges uniformly with respect to the singular perturbation parameter. To obtain an efficient numerical algorithm, such initial value problems are discretized by using appropriate time integrators; here, we have chosen the Implicit Euler method for doing this, obtaining an unconditional and uniformly convergent numerical method in a simple way. Finally, some numerical results are shown in order to illustrate the efficiency of the numerical methods, according to the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper the numerical approximation of 1D parabolic singularly perturbed systems with two equations of reaction-diffusion type is considered. These problems typically exhibit two overlapping boundary layers at both end points of the spatial domain. A decomposition of the exact solution into its regular and singular part is established, given appropriate bounds for the partial derivatives of the exact solution up to sixth order. These bounds are crucial to prove the uniform convergence of a numerical method that combines the classical backward Euler method and a hybrid finite difference scheme defined on a special nonuniform mesh condensing in the layer regions. The numerical method is uniformly convergent in the discrete maximum norm, and it has first and third order of convergence in time and space, respectively. Numerical results for some test problems are showed, illustrating in practice the order of convergence theoretically proved. [ABSTRACT FROM AUTHOR]

Abstract: In this paper 1D parabolic systems of two singularly perturbed equations of reaction–diffusion type are examined. For the time discretization we consider two additive (or splitting) schemes defined on a uniform mesh and for the space discretization we use the classical central difference approximation defined on a Shishkin mesh. The uniform convergence of both the semidiscrete and the fully discrete problems is proved. The additive schemes are used to solve a test problem, and the results obtained with these schemes and the standard discretization using the backward Euler method are compared. Also, numerical results are presented in the case of systems of three equations. All the numerical results show the advantage in computational cost of the additive schemes compared to the standard discretization. [Copyright &y& Elsevier]

Abstract: This paper deals with the numerical approximation of the solution of 1D parabolic singularly perturbed problems of reaction–diffusion type. The numerical method combines the standard implicit Euler method on a uniform mesh to discretize in time and a HODIE compact fourth order finite difference scheme to discretize in space, which is defined on a priori special meshes condensing the grid points in the boundary layer regions. The method is uniformly convergent having first order in time and almost fourth order in space. The analysis of the uniform convergence is made in two steps, splitting the contribution to the error from the time and the space discretization. Although this idea has been previously used to prove the uniform convergence for parabolic singularly perturbed problems, here the proof is based on a new study of the asymptotic behavior of the exact solution of the semidiscrete problems obtained after the time discretization by using the Euler method. Some numerical results are given corroborating in practice the theoretical results. [Copyright &y& Elsevier]

Abstract: In this paper a computational technique is proposed for obtaining a higher order global solution and global normalized flux of singularly perturbed reaction–diffusion two-point boundary-value problems. The HOC (higher order compact) finite difference scheme developed in Gracia et al. (2001) and which is constructed on an appropriate piecewise uniform Shishkin mesh, has been considered to find an almost fourth order convergent solution at mesh points. Using these values, piecewise cubic interpolants based approximations for solution and normalized flux in whole domain have been defined. It has been shown that the global solution and the global normalized flux are also uniformly convergent. Moreover, for the global solution, the order of uniform convergence in the whole domain is optimal, i.e., it is the same as this one obtained at mesh points, whereas, for the global normalized flux, the uniform convergence is almost third order, except at midpoints of the mesh, where it is also almost fourth order. Theoretical error bounds have been provided along with some numerical examples, which corroborate the efficiency of the proposed technique to find good approximations to the global solution and the global normalized flux. [Copyright &y& Elsevier]

Abstract: In this work we are interested in the numerical approximation of 1D parabolic singularly perturbed problems of reaction–diffusion type. To approximate the multiscale solution of this problem we use a numerical scheme combining the classical backward Euler method and central differencing. The scheme is defined on some special meshes which are the tensor product of a uniform mesh in time and a special mesh in space, condensing the mesh points in the boundary layer regions. In this paper three different meshes of Shishkin, Bahkvalov and Vulanovic type are used, proving the uniform convergence with respect to the diffusion parameter. The analysis of the uniform convergence is based on a new study of the asymptotic behavior of the solution of the semidiscrete problems, which are obtained after the time discretization by the Euler method. Some numerical results are showed corroborating in practice the theoretical results on the uniform convergence and the order of the method. [Copyright &y& Elsevier]

Abstract: The growth mode, magnetic and magneto-optical properties of epitaxial Au/Co/Au(111) ultrathin trilayers grown by pulsed-laser deposition (PLD) under ultra-high vacuum are presented. Sapphire wafers buffered with a single-crystalline Mo(110) buffer layer were used as substrates. Owing to PLD-induced interfacial intermixing at the lower Co/Au(111) interface, a close-to layer-by-layer growth mode is promoted. Surprisingly, despite this intermixing, ferromagnetic behavior is found at room temperature for coverings starting at 1 atomic layer (AL). The films display perpendicular magnetization with anisotropy constants reduced by 50% compared to TD-grown or electrodeposited films, and with a coercivity more than one order of magnitude lower . The magneto-optical (MO) response in the low Co thickness range is dominated by Au/Co interface contributions. For thicknesses starting at 3 AL Co, the MO response has a linear dependence with the Co thickness, indicative of a continuous-film-like MO behavior. [Copyright &y& Elsevier]

Abstract: The objective of this paper is to construct some high order uniform numerical methods to solve linear reaction–diffusion singularly perturbed problems. First, for 1D elliptic problems, based on the central finite difference scheme, a new HODIE method is defined on a piecewise uniform Shishkin mesh. Using this HODIE scheme jointly with a two stage SDIRK method, we solve a 1D parabolic singularly perturbed problem. In both cases we prove that the methods are third-order uniform convergent in the maximum norm. Finally, for a 2D parabolic problem of the same type, we show numerically that the combination of the HODIE scheme with a fractional step RK method gives again a third-order uniform convergent scheme. [Copyright &y& Elsevier]

In this paper we construct a numerical method to solve one-dimensional time-dependent convection–diffusion problem with dominating convection term. We use the classical Euler implicit method for the time discretization and the simple upwind scheme on a special nonuniform mesh for the spatial discretization. We show that the resulting method is uniformly convergent with respect to the diffusion parameter. The main lines for the analysis of the uniform convergence carried out here can be used for the study of more general singular perturbation problems and also for more complicated numerical schemes. The numerical results show that, in practice, some of the theoretical compatibility conditions seem not necessary. [Copyright &y& Elsevier]