555 research outputs found

    A splitting uniformly convergent method for one-dimensional parabolic singularly perturbed convection-diffusion systems

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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

    An efficient numerical method for singularly perturbed time dependent parabolic 2D convection–diffusion systems

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    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

    Robust Numerical Methods for Singularly Perturbed Differential Equations--Supplements

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    The second edition of the book "Roos, Stynes, Tobiska -- Robust Numerical Methods for Singularly Perturbed Differential Equations" appeared many years ago and was for many years a reliable guide into the world of numerical methods for singularly perturbed problems. Since then many new results came into the game, we present some selected ones and the related sources.Comment: arXiv admin note: text overlap with arXiv:1909.0827

    On supraconvergence phenomenon for second order centered finite differences on non-uniform grids

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    In the present study we consider an example of a boundary value problem for a simple second order ordinary differential equation, which may exhibit a boundary layer phenomenon. We show that usual central finite differences, which are second order accurate on a uniform grid, can be substantially upgraded to the fourth order by a suitable choice of the underlying non-uniform grid. This example is quite pedagogical and may give some ideas for more complex problems.Comment: 26 pages, 2 figures, 2 tables, 37 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

    Numerical solution of singularly perturbed 2-D convection-diffusion elliptic interface PDEs with Robin-type boundary conditions

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    We consider a singularly perturbed two-dimensional convection-diffusion elliptic interface problem with Robin boundary conditions, where the source term is a discontinuous function. The coefficient of the highest-order terms in the differential equation and in the boundary conditions, denoted by ε, is a positive parameter which can be arbitrarily small. Due to the discontinuity in the source term and the presence of the diffusion parameter, the solutions to such problems have, in general, boundary, corner and weak-interior layers. In this work, a numerical approach is carried out using a finite-difference technique defined on an appropriated layer-adapted piecewise uniform Shishkin mesh to provide a good estimate of the error. We show some numerical results which corroborate in practice that these results are sharp

    An almost third order finite difference scheme for singularly perturbed reaction–diffusion systems

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    AbstractThis 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
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