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

Análisis de las infraestructuras de alumbrado público, sistema hidráulico y del estado de la pavimentación de las vías de la localidad de Albalate del Arzobispo

La falta de documentación sobre las infraestructuras del municipio de Albalate del Arzobispo y del estado de estas, anima a la realización de un proyecto que las recoja de la forma más precisa posible. Este proyecto persigue el objetivo de satisfacer las necesidades de la localidad y a su vez, servir como modelo estándar para futuros estudios similares que puedan ser demandados por empresas privadas o municipios. El objetivo final se basa en obtener una serie de documentos como planos, tablas, etc. que permitan conocer el estado actual de las infraestructuras y por otro lado, sugerir una serie de consejos y propuestas para mejorar dichas infraestructuras. La metodología seguida se basa en una exhaustiva recogida de información sobre el terreno para crear después unos planos precisos y finalmente, analizar y realizar cálculos sobre ellos. El proyecto se organiza según las distintas infraestructuras analizadas: Alumbrado Público, Sistema Hidráulico y Pavimentación de las Calles. Para cada red se crean mapas en AUTOCAD correctamente escalados para posteriormente ser estudiados. Con esto, se cubre la necesidad primordial del municipio de conocer y tener documentos de las infraestructuras actuales. Una vez configurados los planos se procede al estudio y al análisis de estos. Se recogen los datos en tablas que muestran las características más importantes de estas redes, como la longitud total, y los porcentajes de los distintos tipos de materiales, etc. La mayor diferencia se da en el Alumbrado Público, en el cual se lleva a cabo un cálculo aproximado de los diferentes cuadros eléctricos que lo componen utilizando el software de instalaciones eléctricas DMELECT. Finalmente, con todos los datos obtenidos, se sugieren una serie de consejos y propuestas para mejorar las infraestructuras y marcar las pautas de actuación en un futuro. El proyecto consta de dos documentos impresos en papel. La memoria, que es el principal y otro documento denominado anexos que recoge los cálculos y las tablas. Y un tercer documento digital, un CD que recoge los mapas en formato AUTOCAD

Efficient resolution of singularly perturbed coupled systems: Equations of reaction-diffusion

In this communication we consider a class of singularly perturbed linear system of reaction-diffusion type coupled in the reaction terms. To approximate its solution, in [3] J.L. Gracia, F. Lisbona, A uniformly convergent scheme for a system of reaction–diffusion equations, To appear in J. Comp. Appl. Math. the backward Euler method and the central difference scheme on a layer–adapted mesh of Shishkin type was used. We propose a new semi-implicit method which decouples the linear system to be solved at each time level and we prove that it is a uniformly convergent scheme (with respect to the diffusion parameters) in the discrete maximum norm. We display some numerical experiments illustrating in practice the theoretical results. From these examples we can see both the uniform convergence of the numerical method and also its efficiency to approximate the solution of the reaction–diffusion system

Schemes Convergent ε-Uniformly for Parabolic Singularly Perturbed Problems with a Degenerating Convective Term and a Discontinuous Source

We consider the numerical approximation of a 1D singularly perturbed convection-diffusion problem with a multiply degenerating convective term, for which the order of degeneracy is 2p + 1, p is an integer with p ≥ 1, and such that the convective flux is directed into the domain. The solution exhibits an interior layer at the degeneration point if the source term is also a discontinuous function at this point. We give appropriate bounds for the derivatives of the exact solution of the continuous problem, showing its asymptotic behavior with respect to the perturbation parameter ε, which is the diffusion coefficient. We construct a monotone finite difference scheme combining the implicit Euler method, on a uniform mesh, to discretize in time, and the upwind finite difference scheme, constructed on a piecewise uniform Shishkin mesh condensing in a neighborhood of the interior layer region, to discretize in space. We prove that the method is convergent uniformly with respect to the parameter ε, i.e., ε-uniformly convergent, having first order convergence in time and almost first order in space. Some numerical results corroborating the theoretical results are showed

A HODIE method for 2D parabolic problems of convection difusion type

In this work we construct a numerical method to solve a two dimensional convection-diffusion parabolic problem for which the diffusion term can be very small. To deduce the method we use the Peaceman-Rachford scheme to discretize in time and a finite difference scheme of HODIE type, defined on a piecewise uniform Shihskin mesh, for the spatial discretization. The numerical results show that the method is uniformly convergent with respect to the diffusion parameter, having order two in both time and spatial variables. Therefore, the method is more efficient that the schemes used so far to solve this type of problems

Richardson extrapolation on generalized Shishkin meshes for singularly perturbed problems

In this work we are interested in to apply the Richardson extrapolation technique on a type of finite difference schemes, which are used to solve 1D singularly perturbed problems of convection-diffusion type. The numerical method is constructed on generalized Shishkin meshes, which are defined by using a generating function; in all cases the mesh points are condensed in the boundary layer region, in order to obtain a good approximation in the maximum norm. We prove that, if the diffusion coefficient is sufficiently small, an appropriate Richardson extrapolation increase the order of uniform convergence associated to the basic finite difference scheme. Some numerical examples permit us to confirm in practice the theoretical results