The Numerical Simulation Of Time Integral/Fractional Nonlinear Schrodinger Equation Based On Modified Parallel SPH Method

As one of the most important nonlinear models in nonlinear physics,the nonlinear Schrodinger equation(NLSE)has been widely used in the fields of nonlinear optics,plasma physics,condensed matter physics,etc.However,in the actual calculation and simulation process,it is very difficult to obtain the analytical solution because of the existence of complex nonlinear terms.At this time,the theoretical solution of the time fractional nonlinear Schrodinger equation(TF-NLSE)will be more difficult to obtain by analytical means.Therefore,in recent years,the numerical research on NLSE or TF-NLSE has attracted extensive attention from many scholars.However,there are many difficulties in implementing random node distribution or complex irregular area problems by grid-based numerical methods.Therefore,as a kind of smooth particle hydrodynamics(SPH)method,which does not depend on grids at all,it has received extensive attention in the field of numerical solutions involving partial differential equations.At present,there is little research on the application of SPH or improved SPH methods to the efficient solution of NLSE or TF-NLSE problemsBased on the above analysis,this thesis first extends the modified SPH method based on Taylor expansion to the solution of the first-order spatial derivative,and secondly adopts a higher-order splitting format for the time derivative term,and lastly introduces the MPI parallel computing technology to improve the calculation efficiency.Then a high-order split modified parallel SPH(HSS-CPSPH)algorithm which can stably and efficiently solve the high-dimensional nonlinear Schrodinger equation is obtained.Finally,combining the above modified SPH method with the time fractional finite difference method(FDM)to get a local refinement SPH method(LRCSPH-FDM)considering that the SPH method is easy to implement the local refinement scheme for complex irregular area TF-NLSE.The main work of this thesis is as follows:(1)The modified SPH algorithm without kernel derivatives is extended to solve the spatial derivative in the nonlinear Schrodinger equation,combined with the high-order split scheme,and the MPI parallel technology is introduced to obtain a high-order split corrected parallel SPH method that can solve the Dirichlet or periodic boundary condition.Then through the problems with analytical solutions,the error convergence of the simulation equation of the method is analyzed(2)When the above method is applied to the simulation of the non-analytic solution problem,the parallel technology is used for the high-dimensional equation to improve the calculation efficiency,and the calculation efficiency is analyzed,which shows that the parallel calculation with increasing number of CPUs can significantly shorten the calculation time;at the same time,the comparison with the results of FDM method shows that the proposed method can efficiently and accurately predict the nonlinear propagation process of solitary waves and the occurrence of singularities(3)The modified SPH method and the FDM based on Caputo time fractional derivative are combined to solve the time fractional nonlinear Schrodinger equation(TF-NLSE).Then a SPH-FDM method based on local refinement scheme(LRCSPH-FDM)is proposed to improve the accuracy and not increase the calculation too much considering that the SPH method is not limited by the grid.By solving examples with analytical solutions,the advantages of local refinement to improve accuracy are verified,and the error convergence of the proposed method is analyzed Subsequently,the simulation of the TF-NLSE problem on complex irregular regions is performed,demonstrating the advantages of the pure meshless method(4)Using the LRCSPH-FDM proposed above to simulate and predict the problem with no analytical solution,the results are in good agreement with the FDM results,illustrating the superiority of the method in capturing the quantum mechanical properties of nonlinear equations.