Orthogonal Collocation Method And Numerical Analysis For Nonlinear Equations

Nonlinear partial differential equation is an important branch of non-linear science,and its numerical solution is an important research direction of current scientific development.At present,the numerical methods for solving nonlinear equations mainly include iteration method,finite difference method,finite element method and spectral method.In this paper,two kinds of nonlinear equations,Burgers equation and BBM equation,are studied by using orthogonal collocation method.The orthogonal collocation method has a higher convergence order.Compared with the finite element method,the orthogonal collocation method does not need to calculate numerical integration,and the coefficient matrix is easy to obtain.Therefore,the orthogonal collocation method has a wide range of applications.The main work of this paper is as follows:(1)Semi-discrete and full-discrete collocation schemes are proposed for Burgers equation and BBM equation.(2)The equivalence between the orthogonal collocation scheme and the corresponding Galerkin scheme and the existence and uniqueness of the numerical solution are proved.(3)By introducing some techniques and lemmas,the difficulties caused by non-linear terms and mixed derivative terms are solved.Finally,the optimal convergence results are obtained.(4)Numerical examples are compiled to verify the results of theoretical analysis and the feasibility and validity of the orthogonal collocation method.