Luis M. Abia

On the blow-up time convergence of semidiscretizations of reaction-diffusion equations

Abstract 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 norm of the initial conditions, are provided. Convergence of the blow-up times of the semidiscrete problems to the theoretical one is proved.

Runge-Kutta methods for age-structured population models

Abstract We present an algorithm by which the discrete numerical solution of a Volterra integral equation (with non-singular kernel) obtained from a classical Newton-Cotes-type approximation of its integral part is corrected in an iterative way. The correction mechanism, which is based upon the use of non-purely polynomial quadrature rules, turns out to be parallelizable. A parallel algorithm which is suited for a small number of processors is presented and has been implemented on a ring of transputers. Its efficiency is illustrated and analyzed by means of three test equations.

Blow-up for semidiscretizations of reaction-diffusion equations

Abstract 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 reported.

Numerical schemes for a size-structured cell population model with equal fission

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 predicted accuracy of the schemes. We also consider the behaviour of the methods with respect to the different discontinuities that appear in the solution to the problem and the stable size distribution. In addition, the numerical schemes are used to study asynchronous exponential growth.

The Euler method in the numerical integration of reaction-diffusion problems with blow-up

Abstract 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 situations: blow-up of the first Fourier coefficient, symmetric solutions and reaction functions with f(0)=0 . Hence, our analysis shows the key role played by the time-stepping strategies.

Numerical study on the proliferation cells fraction of a tumour cord model

A characteristic curves numerical method based on a regular grid is performed for the solution of a kind of boundary-value problem for a hyperbolic first-order integro-partial-differential equation describing the stationary state of a tumour cord. Numerical evidence on the order of convergence is presented. And a study on the function that gives the fraction of cells which reenters into proliferation is developed.

Numerical integration of a hierarchically size-structured population model with contest competition

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.

Approximating the survival probability in finite life-span population models

Abstract 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 the convergence of this approximation assuming suitable conditions in relation with the unbounded mortality rate that will be reasonable in the real applications of this model. The numerical experiments carried out with typical mortality rates corroborate the interest of this method.

Numerical analysis of a cell dwarfism model

Abstract 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 their optimal rates of convergence. Numerical experiments are also reported in order to demonstrate the predicted accuracy of the schemes. Finally, a comparison study on their efficiency is presented.