We consider a finite difference scheme, called Quickest, introduced by Leonard in 1979, for the convection-diffusion equation. Quickest uses an explicit, Leith-type differencing and third-order upwinding on the convective derivatives yielding a four-point scheme. For that reason the method requires careful treatment on the inflow boundary considering the fact that we need to introduce numerical...

The finite difference methods of Godunov, Hyman, Lax and Wendroff (two-step), MacCormack, Rusanov, the upwind scheme, the hybrid scheme of Harten and Zwas, the antidiffusion method of Boris and Book, the artificial compression method of Harten, and Glimm’s method, a random choice method, are discussed. The methods are used to integrate the one-dimensional Eulerian form of the equations of gas d...

in this paper, a high-order and conditionally stable stochastic difference scheme is proposed for the numerical solution of $rm ithat{o}$ stochastic advection diffusion equation with one dimensional white noise process. we applied a finite difference approximation of fourth-order for discretizing space spatial derivative of this equation. the main properties of deterministic difference schemes,...

We construct a nonstandard 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 R. E. 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-ord...

This paper is concerned with the construction and developing of several nonstandard finite difference (NSFD) schemes in matrix form in order to obtain numerical solutions of epidemic models. In particular, we deal with a classical SIR epidemic model and a seasonal model associated with the evolution of the transmission of respiratory syncytial virus RSV in the human population. The first model ...

A positivity-preserving numerical method for nonlinear Black-Scholes models is developed in this paper. The numerical method is based on a nonstandard approximation of the second partial derivative. The scheme is not only unconditionally stable and positive, but also allows us to solve the discrete equation explicitly. Monotone properties are studied in order to avoid unwanted oscillations of t...

A ‎nonstandard finite difference (NSFD) scheme has been constructed and analyzed for a mathematical model that describes Lotka–Volterra food web model‎. ‎This new discrete system has the same stability properties as the continuous model and,‎on the whole‎, ‎it preserves‎the same local asymptotic stability properties‎. ‎Linearized stability theory and Schur–Cohn criteria are used for local asymp...

In this paper, a NonStandard Finite Difference (NSFD) scheme is constructed, which can be used to determine numerical solutions for an epidemic model with vaccination. Here the NSFD method is employed to derive a set of difference equations for the epidemic model with vaccination. We show that difference equations have the same dynamics as the original differential system, such as the positivit...