A splitting Chebyshev collocation method for Schrödinger–Poisson system

We develop a splitting Chebyshev collocation (SCC) method for the time-dependent Schrödinger–Poisson (SP) system arising from theoretical analysis of quantum plasmas. By means of splitting technique in time, the time-dependant SP system is first reduced to uncoupled Schrödinger and Poisson equations at every time step. The space variables in Schrödinger and Poisson equations are next represented by high-order Chebyshev polynomials, and the resulting system are discretized by the spectral collocation method. Finally, matrix diagonalization technique is applied to solve the fully discretized system in one dimension, two dimensions and three dimensions, respectively. The newly proposed method not only achieves spectral accuracy in space but also reduces the computer-memory requirements and the computational time in comparison with conventional solver. Numerical results confirm the spectral accuracy and efficiency of this method, and indicate that the SCC method could be an efficient alternative method for simulating the dynamics of quantum plasmas.