Open Access
A splitting uniformly convergent method for one-dimensional parabolic singularly perturbed convection-diffusion systems
(Elsevier, 2023) Clavero, Carmelo; Jorge Ulecia, Juan Carlos; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Universidad Pública de Navarra / Nafarroako Unibertsitate Publikoa
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.
Open Access
A multi-splitting method to solve 2D parabolic reaction-diffusion singularly perturbed systems
(Elsevier, 2023) Clavero, Carmelo; Jorge Ulecia, Juan Carlos; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
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.
Open Access
An efficient numerical method for singularly perturbed time dependent parabolic 2D convection-diffusion systems
(Elsevier, 2018) Clavero, Carmelo; Jorge Ulecia, Juan Carlos; Ingeniería Matemática e Informática; Matematika eta Informatika Ingeniaritza; Institute of Smart Cities - ISC
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.

Jorge Ulecia, Juan Carlos

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