In this work, we propose several simple but accurate finite difference schemes to approximate the solutions of the nonlinear Fisher equation, which describes an interaction between logistic growth and diffusion process occurring in many biological and chemical phenomena. All schemes are based upon the
time-splitting finite difference approximations.
The operator splitting transforms the original problem into two subproblems: nonlinear
logistic and linear diffusion, each with its own boundary conditions. The diffusion equation is solved by
three well-known stable and consistent methods while the logistic equation by a combination of method of lagging and a two-step approximation that is not only preserve positivity but also boundedness.
The new proposed schemes and the previous standard schemes are tested
on a range of problems with analytical solutions. A comparison shows
that the new schemes are simple, effective and very successful in solving the Fisher equation.