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
An efficient numerical method for 1D singularly perturbed parabolic convection-diffusion systems with repulsive interior turning points
(Elsevier, 2025-12-15) Clavero, Carmelo; Jorge Ulecia, Juan Carlos; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
In this work, we propose and study a numerical method to solve efficiently one-dimensional parabolic singularly perturbed systems of convection-diffusion type, for which the convection coefficient is zero at an interior point of the spatial domain. We focus our attention on the case of having the same diffusion parameter in both equations; as well we assume adequate signs on the convective coefficients in order to the interior turning point is of repulsive type. Under these conditions, if the data of the problem are composed by continuous functions, the exact evolutionary solution, in general, has regular boundary layers at the end points of the spatial domain. To solve this type of problems, we combine the fractional implicit Euler method and the classical upwind scheme, defined on a special mesh of Shishkin type. The resulting numerical method reach uniform convergence of first order in time and almost first order in space. Numerical results obtained for different test problems are shown which corroborate in practice the uniform convergence of the numerical algorithm and also their computational efficiency in comparison with classical numerical methods used for the same type of problems.
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
A uniformly convergent scheme to solve two-dimensional parabolic singularly perturbed systems of reaction-diffusion type with multiple diffusion parameters
(John Wiley & Sons Ltd, 2021) Clavero, Carmelo; Jorge Ulecia, Juan Carlos; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
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.
Open Access
An efficient uniformly convergent method for multi-scaled two dimensional parabolic singularly perturbed systems of convection-diffusion type
(Elsevier, 2025-01-01) 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 work we solve initial-boundary value problems associated to coupled 2D parabolic singularly perturbed systems of convection-diffusion type. The analysis is focused on the cases where the diffusion parameters are small, distinct and also they may have different order of magnitude. In such cases, overlapping regular boundary layers appear at the outflow boundary of the spatial domain. The fully discrete scheme combines the classical upwind scheme defined on an appropriate Shishkin mesh to discretize the spatial variables, and the fractional implicit Euler method joins to a decomposition of the difference operator in directions and components to integrate in time. We prove that the resulting method is uniformly convergent of first order in time and of almost first order in space. Moreover, as only small tridiagonal linear systems must be solved to advance in time, the computational cost of our method is remarkably smaller than the corresponding ones to other implicit methods considered in the previous literature for the same type of problems. The numerical results, obtained for some test problems, corroborate in practice the good behavior and the advantages of the algorithm.
Open Access
An efficient and uniformly convergent scheme for one-dimensional parabolic singularly perturbed semilinear systems of reaction-diffusion type
(Springer, 2020) Clavero, Carmelo; Jorge Ulecia, Juan Carlos; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
Open Access
A linearly implicit splitting method for solving time dependent semilinear reaction-diffusion systems
(Springer, 2020) Clavero, Carmelo; Jorge Ulecia, Juan Carlos; Estadística, Informática y Matemáticas; Estatistika, Informatika eta Matematika; Institute of Smart Cities - ISC
In this paper we deal with the efficient resolution of a coupled system of two one dimensional, time dependent, semilinear parabolic singularly perturbed partial differential equations of reaction-diffusion type, with distinct diffusion parameters which may have different orders of magnitude. The numerical method is based on a linearized version of the fractional implicit Euler method, which avoids the use of iterative methods, and a splitting by components to discretize in time; so, only tridiagonal linear systems are involved in the time integration process. Consequently, the computational cost of the proposed method is lower than classical schemes used for the same type of problems. The solution of this singularly perturbed problem features layers, what are resolved on an appropriate piecewise uniform mesh of Shishkin type. We show that the method is uniformly convergent of first order in time and of almost second order in space. Numerical results are presented to corroborate the theoretical results.
Embargo
New fractional step Runge-Kutta-Nyström methods up to order three
(Elsevier, 2020) Bujanda Cirauqui, Blanca; Moreta, M. Jesús; Jorge Ulecia, Juan Carlos; Estatistika, Informatika eta Matematika; Institute for Advanced Materials and Mathematics - INAMAT2; Institute of Smart Cities - ISC; Estadística, Informática y Matemáticas
Fractional Step Runge–Kutta–Nyströ (FSRKN) methods have been revealed to be an excellent option to integrate numerically many multidimensional evolution models governed by second order in time partial differential equations. These methods, combined with suitable spatial discretizations, lead to strong computational cost reductions respect to many classical implicit time integrators. In this paper, we present the construction process of several implicit FSRKN methods of two and three levels which attain orders up to three and satisfy adequate stability properties. We have also performed some numerical experiments in order to show the unconditionally convergent behavior of these schemes as well as their computational advantages.