The dynamics of prey–predator interactions are often modeled using differential or difference equations. In this paper, we investigate the dynamical behavior a two-dimensional discrete system. model is formulated in terms equations and derived by nonstandard finite scheme (NSFD), which takes into consideration non-overlapping generations. existence fixed points as well their local asymptotic st...

This thesis concerns M-finite groups and a notion of discrete measure in models of Peano Arithmetic. First we look at a measure construction for arbitrary non-M-finite sets via suprema and infima of appropriate M-finite sets. The basic properties of the measures are covered, along with non-measurable sets and the use of end-extensions. Next we look at nonstandard finite permutations, introducin...

This paper presents a simple and computationally-efficient algorithm for solving steady two-dimensional subsonic and transonic compressible flow over an airfoil. This work uses an interactive viscous-inviscid solution by incorporating the viscous effects in a thin shear-layer. Boundary-layer approximation reduces the Navier-Stokes equations to a parabolic set of coupled, non-linear partial diff...

In this work, a novel second-order nonstandard finite difference (NSFD) method that preserves simultaneously the positivity and local asymptotic stability of one-dimensional autonomous dynamical systems is introduced analyzed. This based on non-local approximations for right-hand side functions differential equations in combination with denominator functions. The obtained results not only resol...

This paper discusses several aspects related to the consistency and stability of the discrete adjoints of upwind numerical schemes. First and third order upwind discretizations of the one-dimensional advection equation are considered in both the finite difference and finite volume formulations. We show that the discrete adjoints may lose consistency and stability near the points where upwinding...

We prove the convergence of a semi-implicit monotone finite difference scheme approximating an initial-boundary value problem for a spatially one-dimensional quasilinear strongly degenerate parabolic equation, which is supplied with two different inhomogeneous flux-type boundary conditions. This problem arises in the modeling of the sedimentation-consolidation process. We formulate the definiti...

We discuss numerical stabilisation of dynamics governed by nonlinear hyperbolic conservation laws through feedback boundary conditions. Using a discrete Lyapunov function we prove exponential decay of the discrete solution to first– order finite volume schemes in conservative form. Decay rates are established for a large class of finite volume schemes including the Lax–Friedrichs scheme. Theore...

We develop new high-order accurate upwind schemes for the wave equation in second-order form. These schemes are developed directly for the equations in second-order form, as opposed to transforming the equations to a first-order hyperbolic system. The schemes are based on the solution to a local Riemann-type problem that uses d’Alembert’s exact solution. We construct conservative finite differe...

We consider the approximation by multidimensional finite volume schemes of the transport of an initial measure by a Lipschitz flow. We first consider a scheme defined via characteristics, and we prove the convergence to the continuous solution, as the time-step and the ratio of the space step to the time-step tend to zero. We then consider a second finite volume scheme, obtained from the first ...