Non-standard Numerical Scheme Research Articles

A second-order nonstandard finite difference method for a general Rosenzweig–MacArthur predator–prey model

In this paper, we consider a general Rosenzweig–MacArthur predator–prey model with logistic intrinsic growth of the prey population. We develop the Mickens’ method to construct a dynamically consistent second-order nonstandard finite difference (NSFD) scheme for the general Rosenzweig–MacArthur predator–prey model. The second-order NSFD method is based on a novel nonlocal approximation using right-hand side function weights and nonstandard denominator functions.Through rigorous mathematical analysis, we show that the NSFD method not only preserves two important and prominent dynamical properties of the continuous-time model, namely positivity and asymptotic stability independent of the values of the step size, but also is convergent of order 2. Therefore, it provides a solution to the contradiction between the dynamic consistency and high-order accuracy of NSFD methods.The proposed NSFD method improves positive and elementary stable nonstandard numerical schemes constructed in a previous work of Dimitrov and Kojouharov (2006). Moreover, the present approach can be extended to construct second-order NSFD methods for some classes of nonlinear dynamical systems.Finally, the theoretical insights and advantages of the constructed NSFD scheme are supported by some illustrative numerical simulations.

A non-standard numerical scheme for an alcohol-abuse model with induced-complications

The prevalence of alcohol-related fatalities worldwide is on the ascendancy not only Ghana, but worldwide. Although the ramifications of alcohol consumption have been the subject of several studies, alcoholism remains a serious concern in public health. This study investigates the dynamics of alcoholism in a population with consumption-induced complications using a deterministic Modelling framework. Using a novel technique, we determined a threshold parameter R0 which we call the basic alcohol-abuse initiation number which is similar to the basic reproduction number for infectious diseases. The model has two mutually-exclusive fixed points whose existence depend on whether or not the R0 is less or greater than unity. Global asymptotic stability of the alcohol-abuse-free fixed point is shown to be associated with R0≤1. Further, forward bifurcation is observed to occur at R0=1, indicating the possibility of eradication of the phenomenon of alcoholism if R0 can be kept below unity over a sufficiently long period of time. Sensitivity analysis also revealed that the probability of initiation into alcohol-abuse by moderate drinkers (β1), followed by the probability of initiation into alcohol-abuse by heavy drinkers (β2) are the most the parameters with the most influence on R0 and consequently on alcohol-abuse persistence. A non-standard finite difference scheme is also developed to numerically simulate the model so as to demonstrate the findings derived from the analysis and also to observe the impact of some epidemiological factors on the dynamics of alcohol-abuse.

An Efficient Non-Standard Numerical Scheme Coupled with a Compact Finite Difference Method to Solve the One-Dimensional Burgers’ Equation

This article proposes a family of non-standard methods coupled with compact finite differences to numerically integrate the non-linear Burgers’ equation. Firstly, a family of non-standard methods is derived to deal with a system of ordinary differential equations (ODEs) arising from the semi-discretization of initial-boundary value partial differential equations (PDEs). Further, a method of this family is considered as a special case and coupled with a fourth-order compact finite difference resulting in a combined numerical scheme to solve initial-boundary value PDEs. The combined scheme has first-order accuracy in time and fourth-order accuracy in space. Some basic characteristics of the scheme are analysed and a section concerning the numerical experiments is presented demonstrating the good performance of the combined numerical scheme.

Extrapolated nonstandard numerical schemes for solving an epidemiological model for computer viruses

Abstract— The aim of this work is to construct high-order numerical schemes that preserve the dynamical properties of a mathematical model describing the spread of computer viruses on the Internet. For this purpose, we first apply Mickens’ methodology to formulate a dynamically consistent nonstandard finite difference (NSFD) scheme for the model under consid­eration. After that, the constructed NSFD scheme is combined with Richardson’s extrapolation method to generate higher-accuracy numerical approximations. The result is that we obtain extrapolated numerical schemes that not only preserve the dynamical proper­ties of the computer virus propagation model but also provide higher-accuracy numerical approximations. In addition, a set of numerical examples is conducted to illustrate and support the theoretical findings and to show the advantages of the proposed numerical schemes.

Theoretical Analysis and Simulation of a Fractional-Order Compartmental Model with Time Delay for the Propagation of Leprosy

This article investigates the propagation of a deadly human disease, namely leprosy. At the outset, the mathematical model is transformed into a fractional-order model by introducing the Caputo differential operator of arbitrary order. A result is established, which ensures the positivity of the fractional-order epidemic model. The stability of the continuous model at different points of equilibria is investigated. The basic reproduction number, R0, is obtained for the leprosy model. It is observed that the leprosy system is locally asymptotically stable at both steady states when R0<1. On the other hand, the fractional-order system is globally asymptotically stable when R0>1. To find the approximate solutions for the continuous epidemic model, a non-standard numerical scheme is constructed. The main features of the non-standard scheme (such as positivity and boundedness of the numerical method) are also confirmed by applying some benchmark results. Simulations and a feasible test example are presented to discern the properties of the numerical method. Our computational results confirm both the analytical and the numerical properties of the finite-difference scheme.

Dynamically consistent nonstandard numerical schemes for solving some computer virus and malware propagation models

This work is devoted to constructing reliable numerical schemes for some computer virus and malware propagation models. We apply the Mickens' methodology to formulate nonstandard finite difference (NSFD) schemes for some epidemiological models describing the spread of computer viruses and malware. Positivity, boundedness and global asymptotic stability (GAS) of the proposed NSFD schemes are studied rigorously. It should be emphasized that the GAS of the NSFD models are established based on an extension of the classical Lyapunov's direct method. As an important consequence, we conclude that the constructed NSFD schemes are dynamically consistent with respect to the positivity, boundedness and GAS of the continuous models. Finally, a set of numerical examples is conducted to support the theoretical findings and to demonstrate advantages of the NSFD schemes over some well-known standard ones. The numerical examples show that the used standard numerical schemes fail to preserve the qualitative dynamical properties of the continuous models for all finite step sizes; consequently, they can generate numerical approximations that are completely different from the exact solutions. Conversely, the NSFD schemes can provide reliable numerical solutions regardless of chosen step sizes.

Local Embedded Discrete Fracture Model (LEDFM)

The study of flow in fractured porous media is a key ingredient for many geoscience applications, such as reservoir management and geothermal energy production. Modelling and simulation of these highly heterogeneous and geometrically complex systems require the adoption of non-standard numerical schemes. The Embedded Discrete Fracture Model (EDFM) is a simple and effective way to account for fractures with coarse and regular grids, but it suffers from some limitations: it assumes a linear pressure distribution around fractures, which holds true only far from the tips and fracture intersections, and it can be employed for highly permeable fractures only. In this paper we propose an improvement of EDFM which aims at overcoming these limitations computing an improved coupling between fractures and the surrounding porous medium by (a) relaxing the linear pressure distribution assumption, (b) accounting for impermeable fractures modifying near-fracture transmissibilities. These results are achieved by solving different types of local problems with a fine conforming grid, and computing new transmissibilities (for connections between fractures and the surrounding porous medium and those through the porous medium itself near to the fractures). Such local problems are inspired from numerical upscaling techniques present in the literature. The new method is called Local Embedded Discrete Fracture Model (LEDFM) and the results obtained from several numerical tests confirm the aforementioned improvements.

Persistence of dynamic consistency of nonstandard numerical schemes for the Fisher-KPP equation

Finite difference schemes for the Fisher-KPP equation are considered that are constructed using non-local discretization of non-linear terms and standard/nonstandard denominators. Their dynamic consistency with respect to positivity, boundedness, steady-states, and diffusion-front propagation are established. Diffusion-fronts are obtained that inflate, deflate, or remain constant at a geometric rate determined by the time-space step size functional relationship. The persistence of said dynamic consistency in terms of their boundedness, positivity, and determining-diffusion curves at small and large scales is rigorously analyzed and numerically supported. Comparisons are presented for three cases using the Euler, nonstandard finite difference, or exact spectral derivative discretization finite difference denominators. Finally, general measures of robustness in persistence are introduced that incorporate (mesh) scale sizes and are used to rank the robustness of the schemes investigated.

A non-standard numerical scheme for an age-of-infection epidemic model

<p style='text-indent:20px;'>We propose a numerical method for approximating integro-differential equations arising in age-of-infection epidemic models. The method is based on a non-standard finite differences approximation of the integral term appearing in the equation. The study of convergence properties and the analysis of the qualitative behavior of the numerical solution show that it preserves all the basic properties of the continuous model with no restrictive conditions on the step-length <inline-formula><tex-math id="M1">\begin{document}$ h $\end{document}</tex-math></inline-formula> of integration and that it recovers the continuous dynamic as <inline-formula><tex-math id="M2">\begin{document}$ h $\end{document}</tex-math></inline-formula> tends to zero.</p>

Reliable approximations for a hepatitis B virus model by nonstandard numerical schemes

In this work, we propose and analyze nonstandard finite difference (NSFD) schemes for an improved hepatitis B virus (HBV) model. Dynamical properties of the constructed NSFD schemes are studied rigorously. It is proven that the NSFD schemes preserve positivity, boundedness and asymptotic stability of the HBV model for all finite step sizes. In addition, convergence and error bounds for the NSFD schemes are also investigated. The theoretical results and advantages of the NSFD schemes over some existing standard finite difference ones are supported and illustrated by a set of numerical examples. The numerical results show that some typical standard numerical schemes cannot preserve the dynamics of the continuous model for some given step sizes; consequently, they generate numerical solutions that are not only unstable but also negative. Conversely, the constructed NSFD schemes preserve the essential mathematical features of the continuous model for any finite step size; therefore, they are suitable to simulate the dynamics of the HBV model over long time intervals.

Mean Square Convergent Non-Standard Numerical Schemes for Linear Random Differential Equations with Delay

In this paper, we are concerned with the construction of numerical schemes for linear random differential equations with discrete delay. For the linear deterministic differential equation with discrete delay, a recent contribution proposed a family of non-standard finite difference (NSFD) methods from an exact numerical scheme on the whole domain. The family of NSFD schemes had increasing order of accuracy, was dynamically consistent, and possessed simple computational properties compared to the exact scheme. In the random setting, when the two equation coefficients are bounded random variables and the initial condition is a regular stochastic process, we prove that the randomized NSFD schemes converge in the mean square (m.s.) sense. M.s. convergence allows for approximating the expectation and the variance of the solution stochastic process. In practice, the NSFD scheme is applied with symbolic inputs, and afterward the statistics are explicitly computed by using the linearity of the expectation. This procedure permits retaining the increasing order of accuracy of the deterministic counterpart. Some numerical examples illustrate the approach. The theoretical m.s. convergence rate is supported numerically, even when the two equation coefficients are unbounded random variables. M.s. dynamic consistency is assessed numerically. A comparison with Euler’s method is performed. Finally, an example dealing with the time evolution of a photosynthetic bacterial population is presented.

On the global asymptotic stability of a hepatitis B epidemic model and its solutions by nonstandard numerical schemes

Very recently, a hepatitis B epidemic model with saturated incidence rate has been proposed and analyzed in Khan et al. (Chaos Solitons Fractals 124:1–9, 2019). The local asymptotic stability of the disease endemic equilibrium (DEE) point of model has been established theoretically but its global asymptotic stability has not been studied. In this paper, we present a mathematically rigorous analysis for the global asymptotic stability of the DEE point of the model. More precisely, we prove that if the DEE point exists, then it is not only locally asymptotically stable but also globally asymptotically stable. Furthermore, we present an alternative proof for the global stability of the disease-free equilibrium point. The main result is that we obtain the complete global stability of the hepatitis B virus model. Besides, we construct nonstandard finite difference (NSFD) schemes preserving the essential qualitative properties of the continuous model. These properties include the positivity of solutions and the stability of the model. Dynamical properties of the proposed schemes are rigorously investigated by mathematical analyses and numerical simulations. Finally, numerical simulations are performed to confirm the validity of the theoretical results as well as the advantages of the NSFD schemes. Numerical simulations also indicate that the NSFD schemes are appropriate and effective to solve the continuous model. Meanwhile, the standard finite difference schemes such as the Euler scheme, the classical fourth-order Runge–Kutta scheme cannot preserve the essential properties of the continuous model. Consequently, they can generate numerical approximations which are completely different from the solutions of the model.

Exact and Nonstandard Finite Difference Schemes for Coupled Linear Delay Differential Systems

In recent works, exact and nonstandard finite difference schemes for scalar first order linear delay differential equations have been proposed. The aim of the present work is to extend these previous results to systems of coupled delay differential equations X ′ ( t ) = A X ( t ) + B X ( t - τ ) , where X is a vector, and A and B are commuting real matrices, in general not simultaneously diagonalizable. Based on a constructive expression for the exact solution of the vector equation, an exact scheme is obtained, and different nonstandard numerical schemes of increasing order are proposed. Dynamic consistency properties of the new nonstandard schemes are illustrated with numerical examples, and proved for a class of methods.

Hopf bifurcation in a grazing system with two delays

Life in the Sahelian zone is very difficult due to lack of resources on a permanent basis. This scarcity of resources leads the pastor, with their flock to organize themselves to better resist the difficult conditions. Pastoralism is of paramount importance for this region. There is an urgent need to understand this way of life in order to help pastors to better organize themselves. In this paper, we propose mathematical models that consist of three food trophic chains. We consider and formalize, in a given location, interactions between browsers, grazers and forage resources. We first analyze the resulting system without delays and in a second time take into account two positive and discrete time-delays. In our model, delays denote an average time required in order that food consumed by browsers and grazers are beneficial for them. Qualitative analysis of the delays-free model shows several equilibria as well as various situations of multistability. Considering delay as parameter, we investigate the effect of delay on the stability of equilibria. It is found that the delays can lead the system dynamic behavior to exhibit stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. By applying the normal form theory and the center manifold theorem, the explicit formula which determines the stability and direction of the bifurcating periodic solutions is determined. Finally based on a nonstandard numerical scheme, we provide some numerical simulations in order to illustrate our qualitative results and support discussions.

Non-standard finite-difference time-domain method for solving the Schrödinger equation

In this paper, an improvement of the finite-difference time-domain (FDTD) method using a non-standard finite-difference scheme for solving the Schrödinger equation is presented. The standard numerical scheme for a second derivative in the spatial domain is replaced by a non-standard numerical scheme. In order to apply the non-standard FDTD (NSFDTD), first, the estimates of eigenenergies of a system are needed and computed by the standard FDTD method. These first eigenenergies are then used by the NSFDTD method to obtain improved eigenenergies. The NSFDTD method can be employed iteratively using the resulting eigenenergies to obtain more accurate results. In this paper, the NSFDTD method is validated using infinite square well, harmonic oscillator and Morse potentials. Significant improvements are found when using the NSFDTD method.

Exact and nonstandard numerical schemes for linear delay differential models

Delay differential models present characteristic dynamical properties that should ideally be preserved when computing numerical approximate solutions. In this work, exact numerical schemes for a general linear delay differential model, as well as for the special case of a pure delay model, are obtained. Based on these exact schemes, a family of nonstandard methods, of increasing order of accuracy and simple computational properties, is proposed. Dynamic consistency of the new nonstandard methods are proved, and illustrated with numerical examples, for asymptotic stability, positive preserving properties, and oscillation behaviour.

A nonstandard numerical scheme for a predator-prey model with Allee effect

A nonstandard numerical scheme for a predator-prey model with Allee effect

Stability Analysis of a Fractional Order Modified Leslie-Gower Model with Additive Allee Effect

We analyze the dynamics of a fractional order modified Leslie-Gower model with Beddington-DeAngelis functional response and additive Allee effect by means of local stability. In this respect, all possible equilibria and their existence conditions are determined and their stability properties are established. We also construct nonstandard numerical schemes based on Grünwald-Letnikov approximation. The constructed scheme is explicit and maintains the positivity of solutions. Using this scheme, we perform some numerical simulations to illustrate the dynamical behavior of the model. It is noticed that the nonstandard Grünwald-Letnikov scheme preserves the dynamical properties of the continuous model, while the classical scheme may fail to maintain those dynamical properties.

A minimalistic model of tree–grass interactions using impulsive differential equations and non-linear feedback functions of grass biomass onto fire-induced tree mortality

Since savannas are important ecosystems around the world, their long term dynamics is an important issue, in particular when perturbations, like fires, occur more or less often. In a previous paper, we developed and studied a tree–grass model that take into account fires as pulse events using impulsive differential equations. In this work, we propose to improve this impulsive model by considering the impact of pulse fire on tree biomass by means of combination of two nonlinear functions of grass and tree biomasses respectively. By considering two impact functions, our model yields more complex dynamics, allowing for the possibility of various bistabilities and periodic solutions, in either grassland or savanna states in the ecosystem. Our mathematical analysis allows extensive and realistic description of savannas ecosystems, than previous modelling approaches. We also highlight several threshold parameters that summarize all possible dynamics, as well as three main parameters of bifurcations in the tree–grass dynamics: the fire period, the tree–grass facilitation/competition parameter, and the fire intensity. Using an appropriate nonstandard numerical scheme, we provide numerical simulations to discuss some ecologically interesting cases that our model is able to exhibit along a rainfall gradient, observable in Central Africa.

Mathematical Analysis of a Size Structured Tree-Grass Competition Model for Savanna Ecosystems

Several continuous-time tree-grass competition models have been developed to study conditions of long-lasting coexistence of trees and grass in savanna ecosystems according to environmental parameters such as climate or fire regime. In those models, fire intensity is a fixed parameter while the relationship between woody plant size and fire-sensitivity is not systematically considered.In this paper, we propose a mathematical model for the tree-grass interaction that takes into account both fire intensity and size-dependent sensitivity. The fire intensity is modeled by an increasing function of grass biomass and fire return time is a function of climate. We carry out a qualitative analysis that highlights ecological thresholds that summarize the dynamics of the system. Finally, we develop a non-standard numerical scheme and present some simulations to illustrate our analytical results.