Some Finite Difference Methods For Solving Fisher-Kolmogorov Equation

In many areas of natural science, there is a lot of phenomenon that can be de-scribed with parabolic equation or equations. Such as some chemical reaction, heatconduction, the transport of various particles,certain biological form and other diffu-sion phenomenon etc. Among that, the Fisher-Kolmogorov(FK) equation model is firstproposedbyR.FisherandKolmogorovin1937,whichisusedtodescribetheinteractionbetween the spread of the population and the accommodation.Nonlinear problem is one of the main research topics in modern mathematics, thisisnotonlyduetotheneedsofthedevelopmentofscienceandtechnology,butalsoduetothe rapid development of computing technology which provides a solution to this prob-lem. It is difficult to deal with the nonlinear terms in the actual numerical experiment,in the numerical experiments, the calculation results of the simplified Newton is inex-actness. Soweusuallyusethemethodbywhichwechangethenonlineartermstolinearterm. Using themethod, we cankeeptheoriginal order ofconvergenceand improvethecomputing speed. Moreover, in the actual application, the scale of the problem is largerin many questions, it is more difficult and time consuming if we use ordinary finite dif-ference method to solve them, so we can deal with the discrete equation with parallelalgorithm, and then we use the numerical method to solve the parabolic partial differ-ential equation problem, and it is important in theoretical significance and applicationvalue.The Classical explicit format has an ideal parallelism,which is very suitable forparallel computing. But since it is conditionally stable, So when it comes to parabolicequation or equations in the large-scale scientific computing, people often use the im-plicit format or Crank-Nicolson format with good stability. However, when solving theimplicit difference scheme, we need to solve the banded equations, and it cannot beapplied on parallel machine directly and effectively. So we can consider how to form the difference scheme to research the parallel algorithm. People developed the blockimplicit structure by Saul’yev asymmetric form, and then set up a variety of explicit-implicit format and pure implicit alternating parallel method by using alternating tech-nology. At last we get the method which has the better stability and the parallelism.Thispapercanbedividedintofourparts. Inthefirstpartwesummarizetheknowl-edge and background which can be used in the paper, then we introduce evolution equa-tion briefly. In particular, to the FK equation, we do a simple summary with the exis-tence and uniqueness of solution and the application of relevant theory. Then, we sum-marize the theory of finite difference method which is used later. Next, we introducethe related knowledge with Classical format, Saul’yev asymmetric form, Alternatingsection C-N format respectively. In the second part, we design a finite difference algo-rithmoftheFKequationwiththeC-Nformat,existence,uniquenessareproved,further,priori estimates and the convergence of difference solutions with order O(τ2+h2) inthe L∞-norm are proved, numerical experimentation agrees with the theory analysis.When structuring the difference scheme, we require to solve nonlinear equations everystep, and the calculation process is rather complicated. In the actual numerical experi-ments, the simplified Newton method is used, but it is not accurate enough. So we willlinearize nonlinear term with the clever way in the third part. After theoretical analysis,on the basis of retaining the original order of convergence, we can simplify the solu-tion of the numerical experiment process. In the fourth part, we use alternating sectionC-N form to deal with the C-N form, then a new parallel format is designed, the theoryprove that the method can keep the original order of convergence, the correspondingerror estimates are given through numerical experiments, and the method can improvecomputing speed.