2-D nonlinear Schroedinger equation. Numerical aspects

Various numerical methods are employed in order to approximate 2-D nonlinear Schroedinger equation, namely: (i) the classical explicit method, (ii) the Crank-Nicolson implicit scheme, (iii) the Hardin-Tappert split step Fourier method, (iv) the operator exponential scheme, (v) the simplified operator exponential scheme. A comparison between these schemes is made. The approach for comparison is to (a) fix the accuracy; (b) leave mesh parameters (Δx,Δy,Δt) free and compare the computing time required to attain such accuracy for various choices of the parameters. The results of our study suggest operator exponential schemes as the most effective numerical schemes for 2-D nonlinear Shroedinger equation.