Error estimates of exponential wave integrator Fourier pseudospectral methods for the nonlinear Schrödinger equation.

In this article, several exponential wave integrator Fourier pseudospectra methods are developed for solving the nonlinear Schrödinger equation in two dimensions. These numerical methods are based on the Fourier pseudospectral discretization for spatial derivative first and then utilizing four types of numerical integration formulas, including the Gautschi-type, trapezoidal, middle rectangle and Simpson rules, to approximate the time integral in phase space. The resulting numerical methods cover two explicit and an implicit schemes and can be implemented effectively thanks to the fast discrete Fourier transform. Additionally, the error estimates of two explicit schemes are established by virtue of the mathematical induction and standard energy method. Finally, extensive numerical results are reported to show the efficiency and accuracy of the proposed new methods and confirm our theoretical analysis.