Split-step finite difference schemes for solving the nonlinear Fisher Equation

In this work, we propose several simple but accurate nite di erence schemes to approximate the solutions of the nonlinear Fisher equation, which describes an interaction between logistic growth and di usion process occurring in many biological and chemical phenomena. All schemes are based upon the time-splitting nite di erence approximations. The operator splitting transforms the original problem into two subproblems: nonlinear logistic and linear di usion, each with its own boundary conditions. The di usion 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, e ective and very successful in solving the Fisher equation.