Juan C. López-Marcos

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.

Galerkin methods for a semilinear parabolic problem with nonlocal boundary conditions

We formulate and analyze a Crank-Nicolson finite element Galerkin method and an algebraically-linear extrapolated Crank-Nicolson method for the numerical solution of a semilinear parabolic problem with nonlocal boundary conditions. For each method, optimal error estimates are derived in the maximum norm.

Numerical schemes for size-structured population equations.

We formulate schemes for the numerical solution of size-dependent population models. Such schemes discretize size by means of a natural grid, which introduces a discrete dynamics. 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.

On the numerical integration of non-local terms for age-structured population models.

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-local terms. The schemes are analyzed and some numerical experiments are also reported in order to show numerically their accuracy.

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 integration of autonomous and non-autonomous non-linear size-structured population models

We present an efficiency study for autonomous and non-autonomous non-linear size-structured population models. The study considers three numerical methods: a characteristics scheme, the Lax-Wendroff method and the box method, which are completely described in the paper. Five test problems are considered with diverse degree of complexity: non-trivial equilibrium, periodic solutions and diverse growth functions. The study of the efficiency takes into account the properties of the numerical schemes (such as the stability) and uses a multiple regression analysis to determine the constants of the leading terms of the corresponding global errors. We show how the tables of errors and cpu-times can be used to explain the meaning of the efficiency results. In addition, we present the convergence analysis of the box method.

Numerical integration of a mathematical model of hematopoietic stem cell dynamics

A mathematical model of hematopoiesis, describing the dynamics of stem cell population, is investigated. This model is represented by a system of two nonlinear age-structured partial differential equations, describing the dynamics of resting and proliferating hematopoietic stem cells. It differs from previous attempts to model the hematopoietic system dynamics by taking into account cell age-dependence of coefficients, that prevents a usual reduction of this system to an unstructured delay differential system. We prove the existence and uniqueness of a solution to our problem, and we investigate the existence of stationary solutions. A numerical scheme adapted to the problem is presented. We show the effectiveness of this numerical technique in the simulation of the dynamics of the solution. Numerical simulations show that long-period oscillations can be obtained in this model, corresponding to a destabilization of the system. These oscillations can be related to observations of some periodical hematological diseases (such as chronic myelogenous leukemia).

An upwind scheme for a nonlinear hyperbolic integro-differential equation with integral boundary condition☆

Abstract We analyze an upwind method for a nonlinear hyperbolic integro-differential equation with an integral boundary condition. The problem considered describes the evolution of the age structure of a population. The analysis is carried out employing an abstract theory of discretizations, based on the notion of stability thresholds and on a result due to Stetter.

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.