A predator-prey model with Beddington-DeAngelis functional response: a non-standard finite-difference method

In this paper, we transform a continuous-time predator-prey model with Beddington–DeAngelis functional response into a discrete-time model by nonstandard finite difference scheme (NSFD). The NSFD model shows complete dynamic consistency with its continuous counterpart for any step size. However, the discrete model of same continuous system obtained by Euler forward method shows dynamic inconsistency for larger step size. Extensive numerical simulations have been done to compare the dynamics of NSFD system and Euler system. Our analysis reveals that dynamics of NSFD model is independent of the step-size, whereas the dynamics of the standard discrete model completely depends on the step-size and produces spurious dynamics like chaos.     

Discretization of an eco-epidemiological model and its dynamic consistency

In this paper, we discretize a continuous-time eco-epidemiological model by non-standard finite difference (NSFD) scheme as well as standard Euler forward scheme. Dynamical properties of both the systems are explored and compared with their continuous-time model. We show that the solution of NSFD system remains positive for all positive initial values. Fixed points and their local stability properties are shown to be identical with that of the continuous model, indicating its dynamic consistency. Dynamics of the Euler model, however, depend on the step–size and therefore dynamically inconsistent. Solutions in this latter method may be negative and allows numerical instabilities, leading to chaos. Extensive numerical simulations have been performed to validate the theoretical results.     

An inverse problem of finding a source parameter in a semilinear parabolic equation

An inverse problem concerning diffusion equation with source control parameter is considered. Several finite-difference schemes are presented for identifying the control parameter. These schemes are based on the classical forward time centred space (FTCS) explicit formula, and the 5-point FTCS explicit method and the classical backward time centred space (BTCS) implicit scheme, and the Crank–Nicolson implicit method. The classical FTCS explicit formula and the 5-point FTCS explicit technique are economical to use, are second-order accurate, but have bounded range of stability. The classical BTCS implicit scheme and the Crank–Nicolson implicit method are unconditionally stable, but these schemes use more central processor (CPU) times than the explicit finite difference mehods. The basis of analysis of the finite difference equations considered here is the modified equivalent partial differential equation approach, developed from the 1974 work of Warming and Hyett. This allows direct and simple comparison of the errors associated with the equations as well as providing a means to develop more accurate finite difference schemes. The results of a numerical experiment are presented, and the accuracy and CPU time needed for this inverse problem are discussed.     

基于非结构自适应网格的复合有限体积法

利用文献[1]中将Lax-Wendroff格式和Lax-Friedrichs格式整体复合作用构成二维无结构网格上的复合型有限体积法,同时利用Delaunay方法,根据流场流动特性变化的梯度值为指示器对网格进行加密和粗化,实现自适应,并将此方法应用到二维浅水波方程的求解上,进行了二维部分溃坝,倾斜水跃的数值实验.结果表明,该方法是一个计算稳定、能适应复杂的求解域、能很好地捕捉激波、且计算速度快的算法.     

A note on an NSFD scheme for a mathematical model of respiratory virus transmission

We construct a non-standard finite difference (NSFD) scheme for an SIRS mathematical model of respiratory virus transmission. This discretization is in full compliance with the NSFD methodology as formulated by Mickens. By use of an exact conservation law satisfied by the SIRS differential equations, we are able to determine the corresponding denominator function for the discrete first-order time derivatives. Our scheme is dynamically consistent with the SIRS differential equations, since the conservation laws are preserved. Furthermore, the scheme is shown to satisfy a positivity condition for its solutions for all values of the time step size.     

Numerical solutions of Burgers–Huxley equation by exact finite difference and NSFD schemes

In this paper, numerical solution of the Burgers–Huxley (BH) equation is presented based on the nonstandard finite difference (NSFD) scheme. At first, two exact finite difference schemes for BH equation obtained. Moreover an NSFD scheme is presented for this equation. The positivity, boundedness and local truncation error of the scheme are discussed. Finally, the numerical results of the proposed method with those of some available methods compared.     

Notes on a finite differencing scheme for the dispersion equation

Finite difference techniques applied to atmospheric dispersion problems often encounter time step limitations due to the variance in the characteristic length scales (horizontal to vertical) of both the field variables and the computational region. Methods to maximize the integration time step are explored and techniques are described which ensure numerical accuracy and stability of these optimized time step techniques.

To circumvent time step limitations arising from consideration of the vertical diffusion term in the dispersion equation, a column implicitization technique is suggested which, through correction terms added to the differencing equation to compensate for truncation errors, provides an efficient and economical atmospheric dispersion solver which is insensitive to the common time step limitations of explicit schemes when large aspect ratio computational volumes are required. Further, it is shown that a relaxed stability criteria proposed by Leonard and Clancy for explicit differencing of the horizontal terms in the dispersion equation, presents a further saving in computational time provided correction terms to the differencing equation are included to eliminate phase and amplitude errors resulting from the larger time steps employed.     

Nonstandard finite difference schemes for a class of generalized convection–diffusion–reaction equations

In this work, a class of nonstandard finite difference (NSFD) schemes are proposed to approximate the solutions of a class of generalized convection–diffusion–reaction equations. First, in the case of no diffusion, two exact finite difference schemes are presented using the method of characteristics. Based on these two exact schemes, a class of exact schemes are presented by introducing a parameter α. Second, since the forms of these exact schemes are so complicated that they are not convenient to use, a class of NSFD schemes are derived from the exact schemes using numerical approximations. It follows that, under certain conditions about denominator function of time‐step sizes, these NSFD schemes are elementary stable and the solutions are positive and bounded. Third, by means of the Mickens' technique of subequations, a new class of implicit NSFD schemes are constructed for the full convection–diffusion–reaction equations. It is shown that, under certain parameters set, these NSFD schemes are capable of preserving the non‐negativity and boundedness of the analytical solutions. Finally, some numerical simulations are provided to verify the validity of our analytical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1288–1309, 2015     

A non-standard finite difference scheme for a diffusive HBV infection model with capsids and time delay

In this paper, a non-standard finite difference (NSFD) scheme for a delayed diffusive hepatitis B virus (HBV) infection model with intracellular HBV DNA-containing capsids is proposed. Dynamic consistency of this NSFD scheme is achieved by showing that the scheme preserves the non-negativity and boundedness of the solutions and the global stability of the homogeneous steady states of the corresponding continuous model without any restriction on spatial and temporal grid sizes. We prove the global stability of the steady states by constructing suitable discrete Lyapunov functions.     

Global stability and a non-standard finite difference scheme for a diffusion driven HBV model with capsids

A diffusion driven model for hepatitis B virus (HBV) infection, taking into account the spatial mobility of both the HBV and the HBV DNA-containing capsids is presented. The global stability for the continuous model is discussed in terms of the basic reproduction number. The analysis is further carried out on a discretized version of the model. Since the standard finite difference (SFD) approximation could potentially lead to numerical instability, it has to be restricted or eliminated through dynamic consistency. The latter is accomplished by using a non-standard finite difference (NSFD) scheme and the global stability properties of the discretized model are studied. The results are numerically illustrated for the dynamics and stability of the various populations in addition to demonstrating the advantages of the usage of NSFD method over the SFD scheme.     

Numerical treatment for nonlinear Brusselator chemical model

Nowadays, numerical models have great importance in every field of science, especially for solving the nonlinear differential equations, partial differential equations, biochemical reactions, etc. The total time evolution of the reactant species which interacts with other species is simulated by the Runge-Kutta of order four (RK4) and by Non-Standard finite difference (NSFD) method. A NSFD model has been constructed for the biochemical reaction problem and numerical experiments are performed for different values of discretization parameter h. The results are compared with the well-known numerical scheme, i.e. RK4. The developed scheme NSFD gives better results than RK4.     

Nonstandard finite difference schemes for Michaelis–Menten type reaction‐diffusion equations

We compare and investigate the performance of the exact scheme of the Michaelis–Menten (M–M) ordinary differential equation with several new nonstandard finite difference (NSFD) schemes that we construct using Mickens' rules. Furthermore, the exact scheme of the M–M equation is used to design several dynamically consistent NSFD schemes for related reaction‐diffusion equations, advection‐reaction equations, and advection‐reaction‐diffusion equations. Numerical simulations that support the theory and demonstrate computationally the power of NSFD schemes are presented. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS

This paper gives the detailed numerical analysis of mixed finite element method for fractional Navier-Stokes equations.The proposed method is based on the mixed finite element method in space and a finite difference scheme in time.The stability analyses of semi-discretization scheme and fully discrete scheme are discussed in detail.Furthermore,We give the convergence analysis for both semidiscrete and flly discrete schemes and then prove that the numerical solution converges the exact one with order O(h2+k),where h and k:respectively denote the space step size and the time step size.Finally,numerical examples are presented to demonstrate the effectiveness of our numerical methods.     

A non‐standard finite difference scheme for an advection‐diffusion‐reaction equation

In this note, a non‐standard finite difference (NSFD) scheme is proposed for an advection‐diffusion‐reaction equation with nonlinear reaction term. We first study the diffusion‐free case of this equation, that is, an advection‐reaction equation. Two exact finite difference schemes are constructed for the advection‐reaction equation by the method of characteristics. As these exact schemes are complicated and are not convenient to use, an NSFD scheme is derived from the exact scheme. Then, the NSFD scheme for the advection‐reaction equation is combined with a finite difference space‐approximation of the diffusion term to provide a NSFD scheme for the advection‐diffusion‐reaction equation. This new scheme could preserve the fixed points, the positivity, and the boundedness of the solution of the original equation. Numerical experiments verify the validity of our analytical results. Copyright © 2014 JohnWiley & Sons, Ltd.     

Exact finite-difference schemes for first order differential equations having three distinct fixed-points

We construct nonstandard finite-difference (NSFD) schemes that provide exact numerical methods for a first-order differential equation having three distinct fixed-points. An explicit, but also nonexact, NSFD scheme is also constructed. It has the feature of preserving the critical properties of the original differential equation such as the positivity of the solutions and the stability behavior of the three fixed-points.     

Analysis of numerical schemes of a mathematical model for sickle cell depolymerization

The purpose of this paper is to present and discuss numerical schemes for a mathematical model that describes carbon monoxide mediated sickle cell polymer melting. Two Runge-Kutta methods are analyzed and shown to be unstable by calculating the first failure value of step size and displaying the bifurcation diagram of RK4. Two nonstandard finite difference (NSFD) schemes are proposed and analyzed; one is shown to be stable subject to a predictable bound on step size, while the second one is unconditionally stable.     

Non-standard finite difference method applied to an initial boundary value problem describing hepatitis B virus infection

In this paper, two non-standard finite difference (NSFD) schemes are proposed for a mathematical model of hepatitis B virus (HBV) infection with spatial dependence. The dynamic properties of the obtained discretized systems are completely analyzed. Relying on the theory of M-matrix, we prove that the proposed NSFD schemes is unconditionally positive. Furthermore, we establish that the NSFD method used preserves all constant steady states of the corresponding continuous initial boundary value problem (IBVP) model. We prove that the conditions for those equilibria to be asymptotically stable are consistent with the continuous IBVP model independently of the numerical grid size. The global asymptotical properties of the HBV-free equilibrium of the proposed NSFD schemes are derived via the construction of a suitable discrete Lyapunov function, and coincides with the continuous system. This confirms that the discretized models are dynamically consistent since they maintain essential properties of the corresponding continuous IBVP model. Finally, numerical simulations are performed from which it is demonstrated that the proposed NSFD method is advantageous over the standard finite difference (SFD) method.     

Nonstandard finite difference method revisited and application to the Ebola virus disease transmission dynamics

We provide effective and practical guidelines on the choice of the complex denominator function of the discrete derivative as well as on the choice of the nonlocal approximation of nonlinear terms in the construction of nonstandard finite difference (NSFD) schemes. Firstly, we construct nonstandard one-stage and two-stage theta methods for a general dynamical system defined by a system of autonomous ordinary differential equations. We provide a sharp condition, which captures the dynamics of the continuous model. We discuss at length how this condition is pivotal in the construction of the complex denominator function. We show that the nonstandard theta methods are elementary stable in the sense that they have exactly the same fixed-points as the continuous model and they preserve their stability, irrespective of the value of the step size. For more complex dynamical systems that are dissipative, we identify a class of nonstandard theta methods that replicate this property. We apply the first part by considering a dynamical system that models the Ebola Virus Disease (EVD). The formulation of the model involves both the fast/direct and slow/indirect transmission routes. Using the specific structure of the EVD model, we show that, apart from the guidelines in the first part, the nonlocal approximation of nonlinear terms is guided by the productive-destructive structure of the model, whereas the choice of the denominator function is based on the conservation laws and the sub-equations that are associated with the model. We construct a NSFD scheme that is dynamically consistent with respect to the properties of the continuous model such as: positivity and boundedness of solutions; local and/or global asymptotic stability of disease-free and endemic equilibrium points; dependence of the severity of the infection on self-protection measures. Throughout the paper, we provide numerical simulations that support the theory.     

Numerical solution for the anisotropic Willmore flow of graphs

The Willmore flow is well known problem from the differential geometry. It minimizes the Willmore functional defined as integral of the mean-curvature square over given manifold. For the graph formulation, we derive modification of the Willmore flow with anisotropic mean curvature. We define the weak solution and we prove an energy equality. We approximate the solution numerically by the complementary finite volume method. To show the stability, we re-formulate the resulting scheme in terms of the finite difference method. By using simple framework of the finite difference method (FDM) we show discrete version of the energy equality. The time discretization is done by the method of lines and the resulting system of ODEs is solved by the Runge–Kutta–Merson solver with adaptive integration step. We also show experimental order of convergence as well as results of the numerical experiments, both for several different anisotropies.     

An optimal 25-point finite difference scheme for the Helmholtz equation with PML

In this paper, we present an optimal 25-point finite difference scheme for solving the Helmholtz equation with perfectly matched layer (PML) in two dimensional domain. Based on minimizing the numerical dispersion, we propose the refined choice strategy for choosing optimal parameters of the 25-point finite difference scheme. Numerical experiments are given to illustrate the improvement of the accuracy and the reduction of the numerical dispersion.