Jean M.-S. Lubuma

Nonstandard finite difference method by nonlocal approximation

Two types of monotonic properties of solutions of differential equations are discussed and general finite difference schemes, which are stable with respect to these properties are investigated. Apart from being elementary stable, these schemes are also shown to preserve qualitative properties of nonhyperbolic fixed points of the differential equations. From the practical point of view, a systematic procedure based on nonlocal approximation, is proposed for the construction of qualitatively stable nonstandard finite difference schemes for the logistic equation, the combustion model and the reaction-diffusion equation.

Non-standard finite-difference methods for vibro-impact problems

Impact oscillators are non-smooth systems with such complex behaviours that their numerical treatment by traditional methods is not always successful. We design non-standard finite-difference schemes in which the intrinsic qualitative parameters of the system—the restitution coefficient, the oscillation frequency and the structure of the nonlinear terms—are suitably incorporated. The schemes obtained are unconditionally stable and replicate a number of important physical properties of the involved oscillator system such as the conservation of energy between two consecutive impact times. Numerical examples, including the Duffing oscillator that develops a chaotic behaviour for some positions of the obstacle, are presented. It is observed that the cpu times of computation are of the same order for both the standard and the non-standard schemes.

Qualitatively stable finite difference schemes for advection-reaction equations

A systematic procedure is proposed and implemented for the design of nonstandard finite difference methods as reliable numerical simulations that preserve significant properties inherent to the solutions of advection-reaction equations. In the case of hyperbolic fixed-points, a renormalization of the denominators of the discrete derivatives is performed for the numerical solutions to display the linear stability properties of the exact solutions. Non-hyperbolic fixed-points are described with the help of two new monotonic properties the construction of schemes, which preserve these properties, being done by nonlocal approximation of nonlinear terms in the reaction terms.

An Improved Theta-method for Systems of Ordinary Differential Equations

The / -method of order 1 or 2 (if / =1/2) is often used for the numerical solution of systems of ordinary differential equations. In the particular case of linear constant coefficient stiff systems the constraint 1/2 h / h 1, which excludes the explicit forward Euler method, is essential for the method to be A -stable. Moreover, unless / =1/2, this method is not elementary stable in the sense that its fixed-points do not display the linear stability properties of the fixed-points of the involved differential equation. We design a non-standard version of the / -method of the same order. We prove a result on the elementary stability of the new method, irrespective of the value of the parameter / ] [0,1]. Some absolute elementary stability properties pertinent to stiffness are discussed.

On nonstandard finite difference schemes in biosciences

We design, analyze and implement nonstandard finite difference (NSFD) schemes for some differential models in biosciences. The NSFD schemes are reliable in three directions. They are topologically dynamically consistent for onedimensional models. They can replicate the global asymptotic stability of the disease-free equilibrium of the MSEIR model in epidemiology whenever the basic reproduction number is less than 1. They preserve the positivity and boundedness property of solutions of advection-reaction and reaction-diffusion equations.

Topological dynamic consistency of non-standard finite difference schemes for dynamical systems

This work expands the mathematical theory which connects continuous dynamical systems and the discrete dynamical systems obtained from the associated numerical schemes. The problem is considered within the setting of Topological Dynamics. The topological dynamic consistency of a family of DDSs and the associated continuous system is defined as topological equivalence between the evolution operator of the continuous system and the set of maps defining the respective DDSs, for all positive time-step sizes. The one-dimensional theory is developed and a few important representative examples are studied in detail. It is found that the design of non-standard topologically dynamically consistent schemes requires some care.

Dirichlet problems in polyhedral domains II: approximation by FEM and BEM

The convergence of the classical finite element method (FEM) and boundary element method (BEM) is poor due to the edge and vertex singularities of the solution of the involved Dirichlet problem relative to an elliptic operator in a polyhedron. Using the global regularity results of Lubuma and Nicaise (1994), we analyse refined FEM and BEM with optimal rates of convergence.

Non-standard methods for singularly perturbed problems possessing oscillatory/layer solutions

We construct and analyze non-standard finite difference methods for a class of singularly perturbed differential equations. The class consists of two types of problems: (i) those having solutions with layer behavior and (ii) those having solutions with oscillatory behavior. Since no fitted mesh method can be designed for the latter type of problems, other special treatment is necessary, which is one of the aims being attained in this paper. The main idea behind the construction of our method is motivated by the modeling rules for non-standard finite difference methods, developed by Mickens. These rules allow one to incorporate the essential physical properties of the differential equations in the numerical schemes so that they provide reliable numerical results. Note that the usual ways of constructing the fitted operator methods need the fitting factor to be incorporated in the standard finite difference scheme and then it is derived by requiring that the scheme be uniformly convergent. The method that we present in this paper is fairly simple as compared to the other approaches. Several numerical examples are given to support the predicted theory.

Stability analysis and dynamics preserving nonstandard finite difference schemes for a malaria model

When both human and mosquito populations vary, forward bifurcation occurs if the basic reproduction number R 0 is less than one in the absence of disease-induced death. When the disease-induced death rate is large enough, R 0 = 1 is a subcritical backward bifurcation point. The domain for the study of the dynamics is reduced to a compact and feasible region, where the system admits a specific algebraic decomposition into infective and non-infected humans and mosquitoes. Stability results are extended and the possibility of backward bifurcation is clarified. A dynamically consistent nonstandard finite difference scheme is designed.

Comparison of Some Standard and Nonstandard Numerical Methods for the MSEIR Epidemiological Model

The paper presents some non‐standard finite difference methods for the MSEIR model. Following a short description of the methods, they are compared with the standard methods usually implemented in scientific software. It is shown that the non‐standard schemes are superior to the standard ones with respect to their qualitative properties and the CPU running time. We show that non‐local approximations of non linear terms in the equations can be of utmost importance to capture the dynamic of the system.