Structure-preserving Algorithms For The Space Fractional Nonlinear Schr(?)dinger Equation

In recent years,with the beautiful depiction of practical problems in many dis-ciplines of fractional differential equations,the theory of fractional calculus and nu-merical algorithm of fractional differential equation have become the focus of many scholars at home and abroad.However,because fractional differential operators are pseudo-differential operators,their non-locality makes it difficult to analyze and cal-culate.The space fractional nonlinear Schr(?)dinger equation is an important kind of fractional differential equation including fractional differential operator.It has been widely used in nonlinear optics,propagation dynamics,water wave dynamics and many other physical problems.Although a lot of work has been done on the numeri-cal methods of the space fractional nonlinear Schr(?)dinger equation for a long time in the past,it is still in the development stage compared with the results of the numer-ical methods of the classical Schr(?)dinger equation.In addition,the construction of the structure-preserving algorithm for conservative systems has been a hot topic of re-search and attention.The main advantage of the structure-preserving algorithm is that it shows more stable and accurate simulation effect than the traditional algorithm in long-time numerical computation.As a Hamiltonian system,space fractional nonlin-ear Schr(?)dinger has two important conserved quantities:energy and mass.Therefore,the development and improvement of corresponding numerical methods and the con-struction of more effective structure preserving schemes are worthy of further study.Before this dissertation,although the conservative Fourier pseudo-spectral meth-ods for solving the space fractional nonlinear Schr(?)dinger equation have been given,there is no work on error analysis.Therefore,in this dissertation,we first estab-lish the optimal L~?convergence estimate of the conservative Fourier pseudo-spectral method.Secondly,we propose several structure-preserving exponential integration methods for the space fractional nonlinear Schr(?)dinger equations,which are linear-ly implicit,fully implicit and explicit,respectively.Among them,the fully implicit method can simultaneously preserve the mass and the modified energy of the space fractional nonlinear Schr(?)dinger equation,while the linearly implicit method and the explicit method preserve the energy and the modified quadratic energy of the discrete system,respectively.In addition,due to the introduction of integrating factor,the fully implicit scheme and the linearly implicit scheme are faster and more accurate than the non-exponential scheme.The whole dissertation contains the following four parts:In Chapter 1,by introducing discrete fractional Sobolev semi-norm and norm,a priori estimate of the numerical solution is given at first.Then,based on the ener-gy method,we establish the optimal L~?error estimates for the conservative Fourier pseudo-spectral method for the space fractional Schr(?)dinger nonlinear equation.In Chapter 2 and Chapter 4,we construct several kinds of structure-preserving exponential integral methods.In Chapter 2,we use the exponential time differencing method in time and the Fourier pseudo-spectral method in space to discretize the space fractional nonlinear Schr(?)dinger equation,and obtain a new energy-preserving linear-ly implicit scheme.Furthermore,by error analysis,we give the optimal unconditional convergence results of the proposed method.In Chapter 3,we construct an arbitrary high-order exponential Runge-Kutta method for solving the space fractional nonlinear Schr(?)dinger equation by using SAV method and exponential integral method.This method can preserve the both mass and energy.In Chapter 4,we propose an explicit energy-preserving exponential time differ-encing method for Hamiltonian partial differential equations.Numerical experiments show the effectiveness of the proposed method in long-time numerical simulation.In Chapter 5,we give the summarization of this dissertation and the prospect of future research.