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In this work we deal with the numerical solution of one dimensional semilinear parabolic singularly perturbed systems of convection-diffusion type. We assume that the coupling in the convection terms is weak and also that the coupling reaction terms are nonlinear. In the case of considering different small diffusion parameters at each equation with different orders of magnitude, the exact solution usually shows overlapping boundary layers on the outflow of the spatial domain. To approximate it, we construct a numerical scheme which combines the upwind finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize in space and a linearized version of the fractional implicit Euler method together with an appropriate splitting by components to discretize in time. Then, the fully discrete method is uniformly convergent with respect to both diffusion parameters, having first order in time and almost first order in space. The choice of this time integrator provokes that only tridiagonal linear systems must be solved at each time step; in this way, the computational cost of the algorithm is considerably lower than this one associated to classical implicit methods. The numerical results obtained for different test problems corroborate in practice the good performance of the numerical algorithm. [ABSTRACT FROM AUTHOR]

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]

Abstract In this paper we deal with solving efficiently 2D linear parabolic singularly perturbed systems of convection–diffusion type. We analyze only the case of a system of two equations where both of them feature the same diffusion parameter. Nevertheless, the method is easily extended to systems with an arbitrary number of equations which have the same diffusion coefficient. The fully discrete numerical method combines the upwind finite difference scheme, to discretize in space, and the fractional implicit Euler method, together with a splitting by directions and components of the reaction–convection–diffusion operator, to discretize in time. Then, if the spatial discretization is defined on an appropriate piecewise uniform Shishkin type mesh, the method is uniformly convergent and it is first order in time and almost first order in space. The use of a fractional step method in combination with the splitting technique to discretize in time, means that only tridiagonal linear systems must be solved at each time level of the discretization. Moreover, we study the order reduction phenomenon associated with the time dependent boundary conditions and we provide a simple way of avoiding it. Some numerical results, which corroborate the theoretical established properties of the method, are shown. [ABSTRACT FROM AUTHOR]

In this paper we consider 1D parabolic singularly perturbed systems of reaction–diffusion type which are coupled in the reaction term. The numerical scheme, used to approximate the exact solution, combines the fractional implicit Euler method and a splitting by components to discretize in time, and the classical central finite differences scheme to discretize in space. The use of the fractional Euler method combined with the splitting by components means that only tridiagonal linear systems must be solved to obtain the numerical solution. For simplicity, the analysis is presented in a complete form only in the case of systems which have two equations, but it can be easily extended to an arbitrary number of equations. If a special nonuniform mesh in space is used, the method is uniformly and unconditionally convergent, having first order in time and almost second order in space. Some numerical results are shown which corroborate in practice the theoretical ones. [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 consider a 1D parabolic singularly perturbed reaction-convection–diffusion problem, which has a small parameter in both the diffusion term (multiplied by the parameter ε 2 ) and the convection term (multiplied by the parameter μ ) in the differential equation ( ε ∈ ( 0 , 1 ] , μ ∈ [ 0 , 1 ] , μ ≤ ε ). Moreover, the convective term degenerates inside the spatial domain, and also the source term has a discontinuity of first kind on the degeneration line. In general, for sufficiently small values of the diffusion and the convection parameters, the exact solution exhibits an interior layer in a neighborhood of the interior degeneration point and also a boundary layer in a neighborhood of both end points of the spatial domain. We study the asymptotic behavior of the exact solution with respect to both parameters and we construct a monotone finite difference scheme, which combines the implicit Euler method, defined on a uniform mesh, to discretize in time, together with the classical upwind finite difference scheme, defined on an appropriate nonuniform mesh of Shishkin type, to discretize in space. The numerical scheme converges in the maximum norm uniformly in ε and μ , having first order in time and almost first order in space. Illustrative numerical results corroborating in practice the theoretical results are showed. [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: A finite difference method for a time-dependent convection–diffusion problem in one space dimension is constructed using a Shishkin mesh. In two recent papers (Clavero et al. (2005) and Mukherjee and Natesan (2009) ), this method has been shown to be convergent, uniformly in the small diffusion parameter, using somewhat elaborate analytical techniques and under a certain mesh restriction. In the present paper, a much simpler argument is used to prove a higher order of convergence (uniformly in the diffusion parameter) than in  and under a slightly less restrictive condition on the mesh. [Copyright &y& Elsevier]

Abstract: This paper addresses the numerical approximation of solutions to coupled systems of singularly perturbed reaction–diffusion problems. In particular a hybrid finite difference scheme of HODIE type is constructed on a piecewise uniform Shishkin mesh. It is proved that the numerical scheme satisfies a discrete maximum principle and also that it is third order (except for a logarithmic factor) uniformly convergent, even for the case in which the diffusion parameter associated with each equation of the system has a different order of magnitude. Numerical examples supporting the theory are given. [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: In this work we define a compact finite difference scheme of positive type to solve a class of 2D reaction–diffusion elliptic singularly perturbed problems. We prove that if the new scheme is constructed on a piecewise uniform mesh of Shishkin type, it provides better approximations than the classical central finite difference scheme. Moreover, the uniform parameter bound of the error shows that the scheme is third order convergent in the maximum norm when the singular perturbation parameter is sufficiently small. Some numerical experiments illustrate in practice the result of convergence proved theoretically. [Copyright &y& Elsevier]

In this work, we design and analyze a finite difference scheme used to solve a class of 2D time-dependent convection–diffusion problems, for which we suppose that the convection term is positive in both spatial directions. We use the Peaceman and Rachford method to discretize in time such problems and high-order differences via an identity expansion finite difference scheme, defined on a piecewise uniform Shishkin mesh, to discretize in space. We prove that the method is uniformly convergent with respect to the diffusion parameter, reaching almost order two in space. A brief discussion concerning the theoretical and practical orders of convergence in time is included, pointing out possible theoretical advances in the future. We present some numerical examples illustrating such a behaviour; they indicate that the numerical method is also suitable in a wider set of singularly perturbed problems than the ones defined by the theoretical restrictions. [ABSTRACT FROM PUBLISHER]

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 work we study a class of HODIE finite difference schemes to solve linear one-dimensional convection–diffusion problems of singular perturbation type. The numerical method is constructed on nonuniform Shishkin type meshes, defined by a generating function, including classical Shishkin meshes and Shishkin–Bakhvalov meshes. We will prove the uniform convergence, with respect to the singular perturbation parameter, of the HODIE scheme on this type of meshes, having order bigger than one. We show some numerical examples confirming in practice the theoretical results and also we see numerically that an appropriate extrapolation will be useful to improve the errors and the order of convergence, when the singular perturbation parameter is sufficiently small. [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]