On the nonstandard finite difference method for reaction-diffusion models

The nonstandard finite difference (NSFD) method is an elegant approach in that it overcomes the numerical instability and bias exhibited by standard finite difference methods for numerically solving nonlinear differ-ential equations. In addition, the NSFD method preserves some qualitative features of the continuous-time model such as boundedness and positivity. But for an important class of models which include diffusion and reaction-diffusion systems that appear in a number of application domains including epidemiology, ecology, and finance, recently introduced NSFD schemes do not guarantee first-order temporal accuracy or consistency. In this paper, we first show this for a reaction-diffusion epidemic model. We then propose an alternative NSFD scheme that guarantees first-order accuracy in time and second-order accuracy in space whilst preserving positivity of the solution. Stability and consistency analyses of the proposed scheme are then presented. To demonstrate the performance of the proposed scheme we show a comparison with the existing NSFD approach for three examples which confirm the superiority of our proposed approach.

Automated summary

The nonstandard finite difference method is a superior approach to numerically solve nonlinear differential equations, as it overcomes the instability and bias of standard finite difference methods. A recent issue with nonstandard finite difference schemes is the lack of first-order temporal accuracy or consistency for important models like diffusion and reaction-diffusion systems. This paper proposes an alternative nonstandard finite difference scheme that guarantees first-order accuracy in time and second-order accuracy in space while preserving solution positivity. Stability and consistency analyses are presented, along with comparisons to existing schemes that confirm the superiority of the proposed approach.