Tesis doctoral de Andrés Arrarás Ventura
This thesis is devoted to the numerical solution of time-dependent partial differential equations of parabolic type. A particular class of such equations governs the so-called reaction-diffusion processes, most commonly arising in environmental modelling and mathematical biology. In this context, we are concerned with either semilinear or quasilinear parabolic problems with mixed derivative terms. Following the method of lines approach, we shall discretize such problems via a two-stage procedure, which sequentially combines the spatial semidiscretization and a time integration method. Initially, we consider the spatial domain to be covered with a logically rectangular mesh. Then, the original problem is discretized in space by using certain generalized finite difference methods, namely: a so-called mimetic scheme and a cell-centered method. Both techniques share two significant features: on one hand, unlike standard finite difference discretizations, they are well fitted to logically rectangular meshes; on the other, they show connections with mixed finite element approximations which incorporate numerical integration. This latter property permits to derive a priori error estimates for the semidiscrete scheme in the abstract setting of galerkin methods. In a second stage, the resulting system of stiff ordinary differential equations is integrated in time by means of a linearly implicit fractional step method. This sort of methods considers a suitable partitioning of the discrete diffusion and source/sink terms into a number of simpler split subterms. Throughout this thesis, we introduce two families of partitioning techniques: a dimensional or component-wise splitting and a so-called domain decomposition splitting. In the former case, we propose an extension of classical alternating direction implicit methods to properly handle mixed derivative terms. The latter improves the efficiency of standard domain decomposition algorithms in the sense of avoiding schwarz iterative procedures. A judicious combination of either of these partitioning functions with a suitable splitting formula gives rise to rather efficient time integrators. A proper choice for this formula is provided by the family of linearly implicit fractional step runge-kutta schemes, which permits to reduce the nonlinear semidiscrete problem to a set of linear systems (one per internal stage) and further allows for parallel implementations. If the original problem is assumed to be quasilinear, a local linearization of the discrete diffusion term is considered before performing the splitting. The convergence analysis of both the semidiscrete and fully discrete schemes is described in detail. Finally, we provide a collection of numerical experiments in order to illustrate the convergence behaviour of the proposed algorithms.
Datos académicos de la tesis doctoral «Mimetic fractional step methods for parabolic problems«
- Título de la tesis: Mimetic fractional step methods for parabolic problems
- Autor: Andrés Arrarás Ventura
- Universidad: Carlos III de Madrid
- Fecha de lectura de la tesis: 25/11/2011
Dirección y tribunal
- Director de la tesis
- Juan Carlos Jorge Ulecia
- Tribunal
- Presidente del tribunal: Luis López bonilla
- Jesús Francisco Palacian subiela (vocal)
- Francisco José Gaspar lorenz (vocal)
- rodolfo Bermejo bermejo (vocal)