Energy Preservation Methods For Nonlinear Schr (?)dinger Equations (NLSEs) With Delta Potentials

Energy-preserving numerical methods are widely used in many scientific areas and have been extensively pursued by many researchers.To the best of our knowledge,however,they are mainly focused on smooth dynamical differential equations.In this thesis,we make an attempt to extending the discussions of energy-preserving methods for non-smooth problems.For simplicity,we are restricted to the model equations,the nonlinear Schr(?)dinger equations(NLSEs)with delta potentials.We reformulate the model equations into an infinite-dimensional Hamiltonian system(IDHS),apply averaged vector field(AVF)methods to the time and obtain the corresponding temporal semi-discrete systems.We prove that the semi-discrete systems possess exact energy preservation properties.As discussed in literature,by using some appropriate distribution theory the semi-discrete systems can be turned into interface problems which admit some non-smooth characters.We then apply suitable immersed interface methods(IIMs)to discretize the space of the semi-discrete systems and get full-discrete systems.The exact preservation of the discrete energy is rigorously proved by using a different representation of the energy as demonstrated in the semi-discrete situation.We remark here that,the choices of representations are the key points in the proofs of the two theorems.In numerical experiments we shows the comparison of average vector field(AVF)and midpoint method.Numerical experiments on attractive solitons are given to validate our theoretical analyses.Together with the normalization preserving and some convergence behaviors,it demonstrates that our methods are considerably good in the long-term calculations.Numerical comparisons with other numerical schemes,exhibit the superiorities of our methods in preserving the energy conservation laws.