New Compact Finite Difference Schemes For The Quantic Nonlinear Schr(?)dinger Equation

In this paper,the initial-boundary value problem of the nonlinear Schr(?)dinger equation with a quintic term is numerically studied by using finite difference method.Two fourth-order compact finite difference schemes for solving the problem are proposed.The optimal error estimates of the numerical solutions are established by using the standard energy method and a "lifting" technique.In the introduction chapter,the physical background and the research progresses of the nonlinear Schr(?)dinger(NLS)equation including the quintic NLS equation is introduced.Two conservation laws including the total mass conservation and total energy conservation of the quantic NLS equation are revisited.Several important inequalities and lemmas used frequently in the paper are arranged in the last part of the chapter.In the second chapter,a linear three-level compact finite difference scheme is derived for solving the quantic NLS equation.The proposed scheme is proved to preserve the total mass and energy in the discrete sense.By introducing the "lifting"technique and using the standard energy method,the optimal error estimate is established without any restriction on the grid ratios.The error bound is proved to be of O(?2+h2)with time step ? and mesh size h.Numerical experiments arw carried out to verify the theoretical results and compared with the existing results in literature.The numerical results show that this scheme has higher computational efficiency while maintaining comparable accuracy.In the third chapter,a linear four-level compact finite difference scheme is derived for solving the quantic NLS equation.The a priori estimation of the solution is given.At the same time,the convergence of the scheme is analyzed and its convergence order is proved to be of O(?2 +h4).Several numerical examples are carried out to test the theoretical results and be compared with the existing results in the references.The numerical results show that the new scheme is not only stable and accurate but also more efficient in the practical computation.The fourth chapter is a summary and prospect part.The two compact finite difference schemes constructed in this paper are summarized from theoretical analysis to numerical experiments,and the future work is prospected.