Analysis And Simulation Of Explicit/Implicit FPM Algorithms For Two Kinks Of Nonlinear Problems

Recently,the nonlinear evolution equation is often used to describe the nonlinear phenomenon in the field of engineering or physical.For example,the nonlinear Cahn-Hilliard(C-H)equation,which describes the phase separation of materials,and the nonlinear time fractional Schrodinger equation(TF-NLSE),which describes the wave phenomenon affected by long term memory in quantum mechanics problems.Therefore,the numerical investigation of nonlinear evolution equation is always one of the research hotspot in computational mathematics or mechanics.Because most nonlinear problems generally include high-order derivatives or nonlinear terms,it is difficult to get their theoretical solutions in an analytical way.Many grid-based numerical methods have been proposed to solve the aforementioned C-H equation and TF-NLSE.However,it is difficult for grid-based methods to deal with these problems high-order derivatives or non-uniform distribution of discrete points.Under such circumstances,Finite Pointset Method(FPM),which is based on Taylor expansion and Weighted Least Squares(WLS),has attracted more attention because of its particle characters.However,FPM has not been extended to solving C-H or TF-NLSE equations.It is difficult to directly apply FPM to C-H equation and TF-NLSE until further modifications of FPM are presentedIn order to improve the stability and accuracy of the FPM to solve the nonlinear problem,this paper first presents an explicit LR-FPM discrete scheme that can accurately solve C-H equation with high-order derivatives.Based on Taylor expansion and the idea of spatial high-order derivative reduction,the implementation of explicit LR-FPM discrete scheme is to continuously apply FPM to discretize the high-order spatial derivatives of C-H equation,to use the second-order precision estimated correction method in the time derivative term,and to combine the advantages of local-refinement.Secondly,a hybrid semi-implicit FPM discrete scheme(H-SIFPM)is proposed for stable and accurate solution of TF-NLSE.In H-SIFPM,FPM is first combined with the finite difference scheme which is based on Caputo fractional derivative,and the implicit discretization of TF-NLSE is combined with Taylor expansion in FPM.Furthermore,the idea of iteration is introduced.Finally,the nonlinear phenomena described by C-H and TF-NLSE problems without analytical solutions are simulated and predicted,and other numerical results are presented for comparison.The main work of this paper is given as follows.(1)To solve C-H equation with high order derivative,we derive a LR-FPM scheme which can accurately solve the C-H equation for the first time,in which the spatial high order derivative is first decomposed into several low order derivatives.Then FPM is applied to the spatial derivative twice and combined with local-refinement technology.The numerical error and convergence of the proposed LR-FPM scheme are analyzed by 1D/2D C-H equations whereas their analytical solutions are available.The numerical results show the proposed LR-FPM method has better stability and higher numerical accuracy.(2)The numerical prediction of C-H problem without analytical solution is carried out using LR-FPM method,and the numerical results are compared with those of Finite Difference Method(FDM).In the simulation,the cases in irregular region are also discussed.It is shown that the proposed method is reliable to predict the phase separation nonlinear diffusion phenomenon described by the C-H equation.Compared with grid-based FDM,FPM has the advantages of flexible popularization and application.(3)In order to solve the TF-NLSE problem based on Caputo fractional derivative,an implicit pure meshless method(H-SIFPM)is proposed by adopting a high-precision implicit discrete scheme in the time fractional derivative,combing the discrete TF-NLSE with the Taylor expansion of FPM,and introducing the iteration.Thus,H-SIFPM can solve TF-NLSE problem stably and accurately.(4)The numerical error and the order of convergence of the proposed H-SIFPM method are investigated by solving constant/variable-order time-fractional NLSE that an analytic solution is available,and the advantages of the proposed method in solving local-refinement and irregular region problems are demonstrated.(5)Using the H-SIFPM predicted the TF-NLSE problem with non-analytical solution,and compared with the results of FDM,the nonlinear dispersion phenomenon in the case of time fractional order is predicted,which is different from the phenomenon in the case of integer order.Numerical results show that the proposed method is reliable for predicting nonlinear dispersion phenomena in time fractional order.