Compact And Conservative Scheme For A Coupled Nonlinear Schr(?)dinger Equations

In this paper, we mainly study initial-boundary valued problem of coupled nonlinear Schro¨di-nger (CNLS) equations, propose two high order compact(HOC) schemes. One has the advantagesof high precision, the other has the advantages of high precision and time-saving. We analyzetheir properties like the existence, convergence and stability, uniqueness of diference solution,then detailed numerical results confirm with our theoretical conclusions.In Chapter1, we firstly introduce some physical background, conserved quantities, presentresearch of CNLS equations. Then some notations and two HOC methods for CNLS equationsare shown. Lastly, we present a family of important lemmas.In Chapter2, we prove the existence of diference solution of HOC scheme I by BrouwerFixed Point Theorem, and discuss convergence and stability, uniqueness of diference solution ofHOC scheme I in detail by discrete energy analysis method. It is confirmed that HOC scheme Ipreserves the discrete charge and energy and the solution of HOC scheme I converges to the exactsolution of original problem with O(h4+τ2) in L2norm.In Chapter3, we also prove the existence of diference solution of HOC scheme II by Leray-Schauder Fixed Point Theorem, and discuss the existence, convergence and stability, uniquenessof diference solution of HOC scheme II by discrete energy analysis method. It is confirmedthat HOC scheme II preserves the discrete charge and energy and the solution of HOC scheme IIconverges to the exact solution of original problem with O(h4+τ2) in L2norm.In Chapter4, we conduct some typical numerical experiments for HOC schemes I and IIto verify the theoretical conclusions, including the accuracy, conservation laws, efciency andvarious physical phenomena described by CNLS equations.