# Luis M. AbiaUniversidad de Valladolid | UVA · Facultad de Ciencias

Luis M. Abia

PhD in Mathematics (University of Valladolid)

## About

33

Publications

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Introduction

**Skills and Expertise**

## Publications

Publications (33)

In this work, we introduce a new semi-Lagrangian numerical method proposed to solve a cell population balance model which describes cell dwarfism, by allowing cell division at any size. We analyze its convergence and derive an optimal rate. Numerical experiments are reported to demonstrate the predicted accuracy of the scheme. Finally, the good beh...

The choice of age as a physiological parameter to structure a population and to describe its dynamics involves the election of the life-span. The analysis of an unbounded life-span age-structured population model is motivated because, not only new models continue to appear in this framework, but also it is required by the study of the asymptotic be...

En este trabajo se presenta el proyecto de innovación docente que se ha llevado a cabo en la asignatura Cálculo Numérico de Primer Curso del Grado en Matemáticas. El objetivo de este proyecto es doble. Por un lado se ha pretendido que el alumnado adquiera varias competencias básicas tanto para el Grado como para su futura vida laboral, como son la...

In this work, we study numerically a model which describes cell dwarfism. It consists in a pure initial value problem for a first order partial differential equation, that can be applied to the description of the evolution of diseases as thalassemia. We design two numerical methods that prevent the use of the characteristic curve x=0, and derive th...

The presence of a steady-state distribution is an important issue in the modelization of cell populations. In this paper, we analyse, from a numerical point of view, the approach to the stable size distribution for a size-structured balance model with an asymmetric division rate. To this end, we introduce a second-order numerical method on the basi...

We consider the numerical approximation of the survival probability in the case of an unbounded mortality rate related to a finite life-span in age-structured population models. Our numerical approach is based on the approximation of the integral that characterizes this probability function by means of an appropriate quadrature rule. We demonstrate...

We formulate schemes for the numerical solution to a hierarchically size-structured population model. The schemes are analysed and optimal rates of convergence are derived. Some numerical experiments are also reported to demonstrate the predicted accuracy of the schemes and to show their behaviour to approaching stable steady states.

We formulate second order finite difference schemes based on Padé rational approximations for the numerical integration of nonlinear age-dependent population models. The schemes are completely analysed and some numerical experiments are also reported in order to show numerically their accuracy.

The proliferative behaviour at the stationary state of a cell population within a tumour cord has been described by a non-classical boundary value problem for a hyperbolic flrst order integro-partial difierential equation (2). The model was theoretically ana- lyzed in (5), where su-cient conditions are given on the fraction of cells which enter pro...

We study the numerical approximation of a size-structured population model whose dependency on the environment is managed by the evolution of a vital resource. We show that this is a difficult task – some numerical methods are not suitable for a long-time integration. We analyze the reasons for the failure.

We study numerically the evolution of a size-structured cell population model, with finite maximum individual size and minimum size for mitosis. We formulate two schemes for the numerical solution of such a model. The schemes are analysed and optimal rates of convergence are derived. Some numerical experiments are also reported to demonstrate the p...

This paper considers the state of the art of the numerical solution of age-structured population models. The different numerical approaches to this kind of problems and the stability and convergence results for them are reviewed. Both characteristic curves methods and finite difference methods are compared with regards to accuracy, efficiency and t...

This paper presents a review of the numerical methods for the solution of the size-structured population balance models. The methods are compared with regards to accuracy, efficiency, generality and mathematical methodology.

A fully discretized scheme, where the Euler method is used for the numerical integration in time, is considered for the approximation of the solutions with blow-up of reaction–diffusion problems. The convergence of the blow-up times of the numerical solutions to the theoretical one is proved, when the time steps are suitable chosen, in three situat...

We formulate explicit second-order finite difference schemes for the numerical integration of non-linear age-dependent population models. These methods have been designed by means of a representation formula for the theoretical solution of the integro-differential equation joint with open quadrature formulae for the numerical approximation of non-l...

Semidiscretizations of reaction-diffusion equations are studied and special attention is devoted to symmetric solutions. Also nonsymmetric solutions are considered when the reaction term is such that f(0) = 0. Sufficient conditions for blow-up in such discretizations are established and upper bounds of the blow-up time, which depend on the maximum...

The behaviour of semidiscretizations of reaction-diffusion equations is studied. Necessary and sufficient conditions for blow-up in such discretizations are given and bounds on the blow-up time are provided. Convergence of the blow-up times of the semidiscrete problems to the theoretical one is established. Also, some numerical experiments are repo...

Difference schemes based on Runge-Kutta methods are introduced for the numerical solution of age-structured population models. The schemes are completely analysed: consistency, stability, existence and convergence are established. Also reported are some numerical experiments in order to show numerically the results proved in our analysis.

Separable Hamiltonian systems of differential equations have the form d{p}/dt = - partial H/partial {q} , d{q}/dt = partial H/partial {p} , with a Hamiltonian function H that satisfies H = T({p}) + V({q}) (T and V are respectively the kinetic and potential energies). We study the integration of these systems by means of partitioned Runge-Kutta meth...

When numerically integrating Hamiltonian systems of differential equations, it is often advantageous to use canonical methods, i.e., methods that preserve the symplectic structure of the phase space, thus reproducing an important feature of the Hamiltonian flow. An s-stage Runge–Kutta (RK) method without redundant stages is canonical if and only if...

A time-discrete pseudospectral algorithm is suggested for the numerical solution of a nonlinear third order equation arising in fluidization. The nonlinear stability and convergence of the new scheme are analyzed. Numerical comparisons with available finite-difference methods are also reported which clearly indicate the superiority of the new schem...

We consider a nonlinear partial differential equation arising in fluidized bed modelling which has been numerically studied by the second author and G. H. Ganser [J. Comput. Phys. 81, No.2, 300-318 (1989; Zbl 0662.76030)]. These authors found that apparently reasonable implicit numerical schemes turn out either to be unconditionally unstable or to...

A continuity argument is employed to prove that the interpolation of the coefficients in nonlinear Galerkin procedures does not result in a reduction of the order of convergence.

Tumour cells growing around blood vessels form cylindrical structures called tumour cords. A nu- merical method is derived for the numerical so- lution of a kind of boundary-value problems for a hyperbolic rst order integro-partial dieren- tial equation describing the stationary state of a tumour cord. Numerical simulations with the method are used...