Barycentric Interpolation Collocation Method For Nonlinear Partial Differential Equations

Mesh less method is a widely used numerical method in recent years.The barycentric interpolation collocation method is a high precision mesh less numerical method.At present,the barycentric interpolation collocation method is mainly used to solve linear partial differential equations,but it is difficult to deal with discrete equations of nonlinear partial differential equations.Furthermore,the thesis constructs direct linearization iteration and Newton-Raphson iteration to discretize the nonlinear elliptic equations and nonlinear parabolic equations based on the barycenter interpolation collocation method,which makes the barycenter interpolation collocation method more universal.By numerical simulation of some nonlinear partial differential equations,the errors of the two new methods in this paper are compared,and the advantages and disadvantages of different methods are verified.Therefore,the research work in this thesis is as follows:Based on the nonlinear elliptic equation and the nonlinear parabolic equation,two iteration methods are applied to two kinds of partial differential equations.Firstly,the discrete equations corresponding to the nonlinear elliptic equations and the nonlinear parabolic equations are constructed by using the barycenter interpolation collocation method.Secondly,the direct linearization iteration schemes and Newton-Raphson obtained by discretizing equation,and the corresponding direct linearization iteration schemes and Newton-Raphson iteration schemes for the discrete equations are written.Thirdly,the boundary conditions are discretized.Finally,the approximate solution of the equation is obtained.The results are as follows:(1)The two iterative methods constructed in this paper can achieve higher computational accuracy in solving nonlinear partial differential equations.(2)Compare the calculation results of two iterative methods,Newton-Raphson iteration method has the high accuracy,high efficiency and unconditional stability in solving nonlinear partial differential equations.(3)Compared with the calculation results of barycentric Lagrange interpolation and barycentric rational interpolation,the barycentric Lagrange interpolation has higher precision and stability,is easy to be used in solving nonlinear partial differential equations.