An explicit nonstandard finite difference scheme for the Allen-Cahn equation

We design explicit nonstandard finite difference schemes for the nonlinear Allen-Cahn reaction diffusion equation in the limit of very small interaction length epsilon. In the proposed scheme, the perturbation parameter is part of the argument of the functional step size, thereby minimizing the restrictions normally associated with standard explicit finite difference schemes. The derivation involves splitting the equation into the space-independent and the time-independent different models. An exact nonstandard scheme is proposed for the space-independent model and energy conservative schemes are proposed for the time-independent model. We show the power of the derived scheme over the existing schemes through several numerical examples.