Unconditional And Optimal Error Estimates Of Finite Difference Schemes For The High-dimensional Klein-Gordon-Schrodinger System And The Gross-Pitaevskii Equations

In this thesis,the nonlinear Klein-Gordon-Schrodinger(KGS)equations and two types of Gross-Pitaevskii(GP)equations are numerically studied by using finite difference method.Firstly,two finite difference schemes are proposed for the KGS equations,and the convergence of the numerical solutions are analyzed.It is proved that the two schemes preserve the total mass and energy in the discrete sense.Besides the standard energy method,an induction argument and a 'lifting' technique are introduced to establish rigorously the optimal H2-error estimates of the numerical solution without imposing any constraints on the grid ratios,while the previous works either are not rigorous enough or often require certain restriction on the grid ratios.Secondly,a conservative finite difference scheme for GP equations with angular momentum rotation is proposed and analyzed.By using the standard energy method and a "lifting" technique as well as "cut-off" technique,the optimal H1-error estimate of the numerical solution is established without any restrictions on the grid ratios.Finally,a linearized,decoupled and fourth-order compact finite difference scheme for the coupled GP equations are proposed.New types of mass functional,magnetization functional and energy functional are defined by using a recursive relation to prove that the new scheme preserves the total mass,magnetization and energy in the discrete sense.The optimal L?-error estimate of the scheme is established without imposing any constraint on the grid ratios.Numerical results are reported to confirm the theoretical analysis for the proposed finite difference schemes.