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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]

The grid approximation of an initial-boundary value problem is considered for a singularly perturbed parabolic convection-diffusion equation with a convective flux directed from the lateral boundary inside the domain in the case when the convective flux degenerates inside the domain and the right-hand side has the first kind discontinuity on the degeneration line. The high-order derivative in the equation is multiplied by ε2, where ε is the perturbation parameter, ε ∊ (0, 1]. For small values of ε, an interior layer appears in a neighbourhood of the set where the right-hand side has the discontinuity. A finite difference scheme based on the standard monotone approximation of the differential equation in the case of uniform grids converges only under the condition N-1 = o(ε), No-1 = o(1), where N +1 and No+1 are the numbers of nodes in the space and time meshes, respectively. A finite difference scheme is constructed on a piecewise-uniform grid condensing in a neighbourhood of the interior layer. The solution of this scheme converges ε- uniformly at the rate O(N-1 ln N +No-1). Numerical experiments confirm the theoretical results. [ABSTRACT FROM AUTHOR]

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]

Develops a numerical method for two-dimensional and time-dependent reaction-diffusion problems. Indications that the method is uniformly convergent with respect to the diffusion parameter; Singular perturbation; Alternating direction.