**PROBLEMAS EVOLUTIVOS EN FISICA E INGENIERIA: TECNICAS DE RESOLUCION ANALITICAS Y NUMERICAS**

MTM2017-83490-P

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**Nombre agencia financiadora**Agencia Estatal de Investigación

**Acrónimo agencia financiadora**AEI

**Programa**Programa Estatal de Fomento de la Investigación Científica y Técnica de Excelencia

**Subprograma**Subprograma Estatal de Generación de Conocimiento

**Convocatoria**Proyectos I+D

**Año convocatoria**2017

**Unidad de gestión**Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016

**Centro beneficiario**UNIVERSIDAD PUBLICA DE NAVARRA

**Identificador persistente**http://dx.doi.org/10.13039/501100011033

## Publicaciones

Found(s) 9 result(s)

Found(s) 1 page(s)

Found(s) 1 page(s)

#### Uniformly convergent additive schemes for 2d singularly perturbed parabolic systems of reaction-diffusion type

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Clavero, C.
- Gracia, J.L.

In this work, we consider parabolic 2D singularly perturbed systems of reaction-diffusion type on a rectangle, in the simplest case that the diffusion parameter is the same for all equations of the system. The solution is approximated on a Shishkin mesh with two splitting or additive methods in time and standard central differences in space. It is proved that they are first-order in time and almost second-order in space uniformly convergent schemes. The additive schemes decouple the components of the vector solution at each time level of the discretization which makes the computation more efficient. Moreover, a multigrid algorithm is used to solve the resulting linear systems. Numerical results for some test problems are showed, which illustrate the theoretical results and the efficiency of the splitting and multigrid techniques.

#### Uniform representations of the incomplete beta function in terms of elementary functions

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Ferreira, Chelo
- López, José L.
- Pérez Sinusía, Ester

We consider the incomplete beta function $B_{z}(a,b)$ in the maximum domain ofanalyticity of its three variables: $a,b,z\\in\\mathbb{C}$, $-a\\notin\\mathbb{N}$,$z\\notin[1,\\infty)$. For $\\Re b\\le 1$ we derive a convergent expansion of$z^{-a}B_{z}(a,b)$ in terms of the function $(1-z)^b$ and of rational functionsof $z$ that is uniformly valid for $z$ in any compact set in$\\mathbb{C}\\setminus[1,\\infty)$. When $-b\\in \\mathbb{N}\\cup\\{0\\}$, the expansionalso contains a logarithmic term of the form $\\log(1-z)$. For $\\Re b\\ge 1$ wederive a convergent expansion of $z^{-a}(1-z)^bB_{z}(a,b)$ in terms of thefunction $(1-z)^b$ and of rational functions of $z$ that is uniformly valid for$z$ in any compact set in the exterior of the circle $\\vert z-1\\vert=r$ forarbitrary $r>0$. The expansions are accompanied by realistic error bounds. Somenumerical experiments show the accuracy of the approximations.

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

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Clavero, C.
- Jorge, J.C.

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.

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

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Clavero, C.
- Jorge, J.C.

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

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Clavero Gracia, Carmelo
- Jorge Ulecia, Juan Carlos

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.

#### New recurrence relations for several classical families of polynomials

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Ferreira, Chelo
- López, José
- Pérez Sinusía, Ester

In this paper, we derive new recurrence relations for the following families of polynomials: Nørlund polynomials, generalized Bernoulli polynomials, generalized Euler polynomials, Bernoulli polynomials of the second kind, Buchholz polynomials, generalized Bessel polynomials and generalized Apostol–Euler polynomials. The recurrence relations are derived from a differential equation of first order and a Cauchy integral representation obtained from the generating function of these polynomials.

**Proyecto:**ES/MINECO/MTM2017-83490-P

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

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Clavero, Carmelo
- Shiromani, Ram
- Shanthi, Vembu

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.

#### A numerical approach for a two-parameter singularly perturbed weakly-coupled system of 2-D elliptic convection–reaction–diffusion PDEs

##### Zaguán. Repositorio Digital de la Universidad de Zaragoza

- Clavero, Carmelo
- Shiromani, Ram
- Shanthi, Vembu

In this work, we consider the numerical approximation of a two dimensional elliptic singularly perturbed weakly-coupled system of convection–reaction–diffusion type, which has two different parameters affecting the diffusion and the convection terms, respectively. The solution of such problems has, in general, exponential boundary layers as well as corner layers. To solve the continuous problem, we construct a numerical method which uses a finite difference scheme defined on an appropriate layer-adapted Bakhvalov–Shishkin mesh. Then, the numerical scheme is a first order uniformly convergent method with respect both convection and diffusion parameters. Numerical results obtained with the algorithm for some test problems are presented, which show the best performance of the proposed method, and they also corroborate in practice the theoretical analysis.

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

##### Academica-e. Repositorio Institucional de la Universidad Pública de Navarra

- Clavero, Carmelo
- 0000-0001-5906-6125

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., This research was partially supported by the project MTM2017-83490-P and the Aragón Government and European Social Fund (group E24-17R ).

**Proyecto:**AEI/Plan Estatal de Investigación Científica y Técnica y de Innovación 2013-2016/MTM2017-83490-P