Non standard finite difference scheme preserving dynamical properties

TL;DR

This paper develops a non-standard finite difference scheme for 2D differential equations, ensuring the preservation of fixed points and stability, with proven convergence and demonstrated advantages over traditional methods.

Contribution

It introduces a novel non-standard finite difference scheme that preserves key dynamical properties of the continuous system, including fixed points and their stability.

Findings

The scheme converges unconditionally.

It preserves fixed points and their stability.

Numerical examples show improved performance over standard methods.

Abstract

We study the construction of a non-standard finite differences numerical scheme for a general class of two dimensional differential equations including several models in population dynamics using the idea of non-local approximation introduced by R. Mickens. We prove the convergence of the scheme, the unconditional, with respect to the discretisation parameter, preservation of the fixed points of the continuous system and the preservation of their stability nature. Several numerical examples are given and comparison with usual numerical scheme (Euler, Runge-Kutta of order 2 or 4) are detailed.