Research On The Mesh-free Method For The Nonlinear Schr?dinger Equation

This dissertation mainly studies the mesh-free method for nonlinear Schr?dinger equations with constant and variable coefficients.As a pure mesh-free method,the smooth particle hydrodynamics(SPH)method has been widely use in the fields of fluid mechanics and solid mechanics.And the traditional SPH method has the problems of insufficient calculation accuracy and poor stability due to the lack of boundary particles.Therefore,based on the idea of Taylor series expansion and symmetric kernel approximation,this dissertation establishes a first-order symmetric SPH method without kernel derivative calculation.Using the relevant conclusions of the Euler-MacLaurin formula and the weighted Monte Carlo integral,the relevant conclusions of the convergence analysis for the first-order symmetric SPH method under the condition of uniform particle spacing and non-uniform particle spacing are obtained.Then the first-order symmetric SPH method is combined with the idea of splitting method,and the method of splitting-step corrective SPH is finally obtained.As a kind of nonlinear dynamic equation,the nonlinear Schr?dinger equation has always been a research hotspot in the field of computational mechanics.Since the exact solution is difficult to be obtained in most cases,the splitting-step corrective SPH method is applied to the nonlinear Schr?dinger equation to obtain the discrete model.Then the relevant conclusions obtained by the convergence analysis are verified by numerical simulation.At the same time,the effects of particle distribution and particle number on the numerical accuracy,numerical convergence and numerical stability of the splitting-step corrective SPH method are studied.In addition,the influence of the smooth function and the smooth length on the numerical accuracy of the splitting-step corrective SPH method is studied by the control variable method.