A multi-splitting method to solve 2D parabolic reaction-diffusion singularly perturbed systems

In this paper we design and analyze a numerical method to solve a type of reaction–diffusion 2D parabolic singularly perturbed systems. The method combines the central finite difference scheme on an appropriate piecewise uniform mesh of Shishkin type to discretize in space, and the fractional implicit Euler method together with a splitting by directions and components of the reaction–diffusion operator to integrate in time. We prove that the method is uniformly convergent of first order in time and almost second order in space. The use of this time integration technique has the advantage that only tridiagonal linear systems must be solved to obtain the numerical solution at each time step; because of this, our method provides a remarkable reduction of computational cost, in comparison with other implicit methods which have been previously proposed for the same type of problems. Full details of the uniform convergence are given only for systems with two equations; nevertheless, our ideas can be easily extended to systems with an arbitrary number of equations as it is shown in the numerical experiences performed. The numerical results show in practice the qualities of our proposal

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

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

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

A uniformly convergent scheme to solve two-dimensional parabolic singularly perturbed systems of reaction-diffusion type with multiple diffusion parameters

In this work, we deal with solving two-dimensional parabolic singularly perturbed systems of reaction-diffusion type where the diffusion parameters at each equation of the system can be small and of different scale. In such case, in general, overlapping boundary layers appear at the boundary of the spatial domain and, because of this, special meshes are required to resolve them. The numerical scheme combines the central difference scheme to discretize in space and the fractional implicit Euler method together with a splitting by components to discretize in time. If the fully discrete scheme is defined on an adequate piecewise uniform Shishkin mesh in space then it is uniformly convergent of first order in time and of almost second order in space. Some numerical results illustrate the theoretical results. © 2020 John Wiley & Sons, Ltd

International Conference on Dynamic Control and Optimization - DCO 2021: book of abstracts

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Finite elements for scalar convection-dominated equations and incompressible flow problems - A never ending story?

The contents of this paper is twofold. First, important recent results concerning finite element methods for convection-dominated problems and incompressible flow problems are described that illustrate the activities in these topics. Second, a number of, in our opinion, important problems in these fields are discussed