Numerical Studies On Some Nonlinear Evolution Partial Differential Equations

In this article,we utilize finite difference,orthogonal spline collocation,time-splitting and spectral mthod to study the numerical solutions of Schr?dinger-Boussinesq（SBq）equations,Klein-Gordon-Schr?dinger（KGS）equations,coupled Gross-Pitaevskii（CGP）equations and modified Gross-Pitaevskii（MGP）equation.In order to construct three points compact finite difference scheme for SBq equations,we equivalently transforme the fourth order partial differential equations（PDE）into a second order PDE by means of order reduction method.Followed by constructed two conservative nonlinear compact finite difference schemes,we establish the priori estimations of the numerical solutions from the discrete energy expression,and the existence,convergence,stability are analyzed via discrete energy methods.The nonlinear comapct finite difference schemes are very time consuming,thus we construct a conservative linear compact finite difference（LCFD）scheme,which could improve the computational efficiency evidently.However,LCFD could not preserve the total energy rigorously,thus we redefine an new sequence,and LCFD is proved to preserve the discrete energy expression which is defined as a recursion relationship,and the convergence and stablity are investigated by means of cut-off truncation function method.Then we utilize orthogonal spline collocation（OSC）method to solve SBq equations,and two conservative OSC schemes are constructed and analyzed.Based on the conservative energy formula,we have proved the priori estimation of the nonlinear OSC scheme,then the existence,convergence and stablity are studied by discrete energy meyhods.Meanwhile,the linear OSC scheme is proved to conserve the total energy which is defined as a recursion relationship,and the convergence and stablity are analyzed by means of cut-off truncation function method.Furthermore,we approximate SBq equations by Fourier pseudo-spectral methods.Therefore,we formulate a time-splitting exponential wave integrator Fourier pseudo-spectral（TS-EWI-FP）method for SBq equations.The main features of TS-EWI-FP method are based on:（I）the applications of a time-splitting Fourier pseudo-spectral method for Schr?dinger-like equation,（II）the utilizations of exponential wave integrator Fourier pseudo-spectral method for Boussinesq-like equation.The scheme is fully explicit and very efficient due to FFT.TS-EWI-FP method is lack of rigorously numerical analysis,thus we design an exponential wave integrator Fourier pseudo-spectral（EWI-FP）method for KGS equations.EWI-FP method is fully explicit,and it is of spectral accuracy in space and second order accuray in time.We have proved the H¹-norm error estimations of EWI-FP method via mathematical induction method.For KGS in high space dimensions,we could obtain theH²-norm error estimations of EWI-FP under stronger regularity conditions.For the coupled Gross-Pitaevskii（CGP）equations,we have investigated and analyzed the explicit finite difference scheme for CGP equations with angular momentum rotation terms.In fact,the explicit difference schem of CGP equations is easy to be designed,thus the motivations of the work is to establish the optimal error estimations of the scheme.We firstly investigate the convergence of the explicit scheme in the ₂L-norm via mathematical induction method,then we establishe the conditional convergence for the errors in the L_?-norm by means of the transformation between the time and space direction and order reduction method.A time-splitting finite difference（TSFD）method for the modified Gross-Pitaevskii（MGP）equation is constructed.TSFD could be realized by fast sine transform,and there is no need to solve large scale linear equations.In addition,the computational complexity of TSFD would not be increased along with the improvement of the spatial accuracy,thus we could design higher compact finente difference to obtain the spectral-like accuracy.TSFD methods with fourth and sixth order accuracy are constructed and investigated.