an:06157401
Zbl 1277.65073
Xu, Yiqiang; Zhang, Luming
Alternating direction implicit method for solving two-dimensional cubic nonlinear Schr??dinger equation
EN
Comput. Phys. Commun. 183, No. 5, 1082-1093 (2012).
00297043
2012
j
65M06 65M12 35Q55
cubic nonlinear Schr??dinger equation; alternating direction implicit; high-order compact; extrapolation technique
Summary: Four alternating direction implicit (ADI) schemes are presented for solving two-dimensional cubic nonlinear Schr??dinger equations. Firstly, we give a Crank-Nicolson ADI scheme and a linearized ADI scheme both with accuracy \(O(\Delta t^2+h^2)\), with the same method, use fourth-order Pad?? compact difference approximation for the spatial discretization; two HOC-ADI schemes with accuracy \(O(\Delta t^2+h^4)\) are given. The two linearized ADI schemes apply extrapolation technique to the real coefficient of the nonlinear term to avoid iterating to solve. Unconditionally stable character is verified by linear Fourier analysis. The solution procedure consists of a number of tridiagonal matrix equations which make the computation cost effective. Numerical experiments are conducted to demonstrate the efficiency and accuracy, and linearized ADI schemes show less computational cost. All schemes given in this paper also can be used for two-dimensional linear Schr??dinger equations.