Accurate Numerical Methods With Analysis Of Several Nonlinear Dispersive Equations

Nonlinear Schr(?)dinger equation,Klein-Gordon-Dirac equation,Klein-Gordon equation with weak nonlinear term and nonlinear Schr(?)dinger equation with wave operator and other nonlinear dispersive equations have very important applications in quantum mechanics,nonlin-ear optics,plasma physics,materials science,biological science and many other fields.In this thesis,finite difference method,Galerkin finite element method and exponential wave integral Fourier pseudo-spectral method are used to study these nonlinear dispersive equations,and a number of new high-precision numerical schemes are designed,some of which can maintain the conservation property or the time symmetry property of the original problems in the dis-crete sense.Mathematical tools such as mathematical induction,cut-off technique and lifting technique are introduced to establish the optimal error estimate of the scheme combined with the energy analysis method.The theoretical results are tested by numerical examples,and the dynamic behavior of the equation is simulated.Firstly,the nonlinear Schr(?)dinger equation with general nonlinear terms and external po-tentials is numerically studied by Galerkin finite element method.A three-level linearized scheme is proposed.The energy analysis method is combined with mathematical induction method and several inverse Sobolev inequalities to establish the optimal L~2-error estimate of the scheme without any requirement on the grid ratio.Compared with the results in the litera-tures,the proof procedure in this thesis is more simple and less required for the regularity of the exact solutions and nonlinear terms.Numerical results verify the error estimate and conserva-tion properties of the scheme.Secondly,the finite difference method is used to numerically study the two-dimensional Klein-Gordon-Dirac equation.In the process of designing and analyzing the scheme,the main difficulty comes from the existence of nonlinear coupling terms.In order to overcome this difficulty,two finite difference schemes of linear decoupling are proposed.It is proved that the schemes maintain the conservation of mass and energy in the discrete sense of the original problem.The optimal error estimate of the scheme is established by using the energy analysis method combined with the lifting technique without any requirement on the grid ratio.The scheme is proved to have second order precision in both spatial and temporal direction.The error estimate and conservation law are verified by numerical examples,and the dynamic behavior of the equation is simulated.Thirdly,this thesis numerically studies the long-time dynamic behavior of the cubic nonlin-ear Klein-Gordon equation with weak nonlinear terms by using Crank Nicolson pseudo-spectral method,and established the optimal error estimate of the scheme.According to the error es-timate,the optimal strategy for grid selection is h≤O(ε~β/r),τ≤O(ε~β/2).In addition,the Klein-Gordon equation with weak nonlinear term can be transformed into a form with high fre-quency oscillation by scaling in the time direction,that is,the solution changes gently in the spatial direction,but there is a high frequency oscillation with wavelength O(ε~β)in the tem-poral direction.The error upper bound and energy conservation properties of the scheme are verified by a large number of numerical results.Finally,the nonlinear Schr(?)dinger equation with wave operator is numerically studied.The main feature of this equation is the existence of high frequency oscillation in the temporal direction,which is mainly caused by the wave operator and the initial velocity.The existence of high frequency oscillations brings essential difficulties to the construction and uniform error estimate of high resolution schemes.In this thesis,two new Fourier pseudo-spectral schemes are constructed for the periodic boundary problem of this equation by means of the exponential wave integral factor method,and the uniform error estimates of the scheme are established by means of the energy analysis method combined with the cut-off technique and some recursive relations for both well-prepared and ill-prepared initial data.Compared with the results in the literature,the proof procedure in this thesis is more simple.The uniform error estimate of the scheme is verified by numerical experiments,and the numerical accuracy and computational efficiency of the two schemes are compared.