Crank-nicolson Method And Precise Integration Method For Solving Fisher Equation

In this paper, the numerical methods for the Fisher equation of theinitial-boundary problem are investigated. As a classic model of reaction diffusionequation, Fisher equation was used to describe nonlinear problem, such as thehydrodynamics, plasma physics, thermonuclear reaction, population growth andspread of infectious diseases, etc. Fisher equation is a kind of important nonlinearequation, it has been widely used in the fields of physics, chemistry, biology, etc. Inmany cases, the solution of Fisher equation is expressed very difficultly with apractical analytic expression. Hence, it is very important to study the numericalsolution of Fisher equations both in theory and applied fields.Firstly, this paper mainly describes the background, the significance and theresearch status of Fisher equation. And we make a brief introduction to the finitedifference method and the precise integration method.Secondly, we study the initial-boundary problem of Fisher equation by usingfinite difference method. We set up a second order implicit scheme for the equation,which has high accuracy and quick convergent ratio. The convergence and stabilityof the difference scheme are proved with the energy analysis method. At last thenumerical experiment shows that the method not only has higher precision, but alsohas good stability, and it has good applicability.Finally, the precise integration methods for solving the Fisher equation aregiven after a series of derivation. Through the theoretical analysis we know that ifwe apply the appropriate numerical methods to the nonlinear term, the accuracy ofprecise integration method can achieve arbitrary order. Andthe algorithm analysisand numerical experiment was carried out. The results demonstrate that theaccuracy is better.