In this paper, a parallel Laplace transform spectral element method for the linear Schr?dinger equation and the cubic nonlinear Schr?dinger equation is introduced. The main advantage of this new scheme is that it has the high accuracy of spectral methods and the parallel efficiency of the Laplace transform method. The key idea is twofold. First, Laplace transform is utilized to eliminate time dependency and the subsequent boundary value problems are discretized by using spectral element method. Second, a parallel Laplace transform method is developed to improve the computational efficiency. For the cubic nonlinear Schr?dinger equation, the increment linearization technique is employed to deal with the nonlinear terms. Numerical experiments are addressed to demonstrate the accuracy and effectiveness of the proposed method. Compared with the Crank–Nicolson scheme, the CPU time is greatly saved by the parallel Laplace transform for both linear and nonlinear problems.
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