On the symmetrization and composition of nonstandard finite difference schemes as an alternative to Richardson's extrapolation

Abstract

From the classical explicit Euler scheme of first order, nonstandard finite difference (NSFD) schemes were envisioned to mimic the essential properties of the governing differential equation model for every time step-size. In the context of compartmental epidemiological models, these properties are generally concerned with the positivity of subpopulations, conservation laws (dynamics of the total population), and stability. However, for autonomous systems, the symmetry (self-adjoint) condition is not preserved. Compartmental epidemiological models are Poisson systems, so methods from geometric numerical integration should be applicable. It is found that symmetrization of NSFD schemes does not respect positivity for every step-size, though other characteristics are maintained and second order is reached. Composition through the Lie formalism can then be applied to obtain higher-order schemes. This is a more efficient and consistent alternative to Richardson's extrapolation, which has often been used to go beyond order one.

Keywords:

• Mickens' nonstandard finite difference scheme
• structure preservation and consistency
• symmetrization
• higher-order integrators via composition
• autonomous compartmental epidemiological model
• Richardson's extrapolation

2020 Mathematics Subject Classifications:

• 65L12
• 65L05
• 92D25

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

Marc Jornet was supported by a postdoctoral contract from Universitat Jaume I, Spain (Acció 3.2 del Pla de Promoció de la Investigació de la Universitat Jaume I per a l'any 2020). Marc Jornet was also funded by project [grant number UJI-B2019-17] from Universitat Jaume I, Spain.