systems in which reaction terms are coupled to diffusion and advection transports arise in awide range of chemical engineering applications, physics, biology and environmental. in these cases, thecomponents of the unknown can denote concentrations or population sizes which represent quantities andthey need to remain positive. classical finite difference schemes may produce numerical drawbacks s...

classical explicit finite difference schemes are unsuitable for the solution of the famous black-scholes partial differential equation, since they impose severe restrictions on the time step. furthermore, they may produce spurious oscillations in the solution. we propose a new scheme that is free of spurious oscillations and guarantees the positivity of the solution for arbitrary stepsizes. the...

Anisotropic diffusion filters with a diffusion tensor are successfully used in many image processing and computer vision applications, ranging from image denoising over compression to optic flow computation. However, finding adequate numerical schemes is difficult: Implementations may suffer from dissipative artifacts, poor approximation of rotation invariance, and they may lack provable stabil...

In this paper, we derive a discretized multi-group epidemic model with time delay by using a nonstandard finite difference (NSFD) scheme. A crucial observation regarding the advantage of the NSFD scheme is that the positivity and boundedness of solutions of the continuous model are preserved. Furthermore, we show that the discrete model has the same equilibria, and the conditions for their stab...

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 asymptoticstab...

When one solves differential equations, modeling physical phenomena, it is of great importance to take physical constraints into account. More precisely, numerical schemes have to be designed such that discrete solutions satisfy the same constraints as exact solutions. Based on general theory for positivity, with an explicit third-order Runge-Kutta method (we will refer to it as RK3 method) pos...

We show how to construct the nonstandard hull of certain infinite-dimensional Lie algebras in order to generalize a theorem of Pestov on the enlargeability of Banach-Lie algebras. In the process, we consider a nonstandard smoothness condition on functions between locally convex spaces to ensure that the induced function between the nonstandard hulls is smooth. We also discuss some conditions on...

in this paper, the reorganization of the denominator of the discrete derivative and nonlocal approximation of nonlinear terms are used in the design of nonstandard finite difference schemes (nsfds). numerical examples confirming then efficiency of schemes, for some differential equations are provided. in order toillustrate the accuracy of the new nsfds, the numerical results are compared with s...