Nonstandard finite difference schemes for linear and non-linear Fokker–Planck equations

The goal of this paper is to develop a nonstandard finite difference-based numerical technique for solving the one-dimensional linear and non-linear Fokker–Planck equations. Characteristics of the nonstandard finite difference method are presented to understand the development of the proposed method. Conditions for the dynamic consistency of positivity and stability of the schemes are obtained. Numerical experiments have been carried out to demonstrate the competitiveness of the proposed methods in comparison to some existing standard methods. In support of the proposed method and analysis, the \(l_2\) and \(l_\infty \) errors are also presented.