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.
Nonlinear Schrödinger and Poisson system Chebyshev collocation method Splitting method Quantum plasmas
Mathematics Subject Classification
35Q55 65Z05 65N12 65N35
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The research of Z. Liang is supported in part by the Natural Science Foundation of China under Grant nos. 11371097, 11571249. The research of H. Wang is supported in part by the Natural Science Foundation of China under Grant no. 91430103.
Anderson D et al (2002) Statistical effects in the multistream model for quantum plasmas. Phys Rev E 65:046417CrossRefGoogle Scholar
Bao W, Mauser NJ, Stimming HP (2003) Effective one particle quantum dynamics of electrons: a numerical study of the Schrödinger–Poisson-\(\Xi \alpha \) model. Comm Math Sci 1:809–831CrossRefzbMATHGoogle Scholar
Canuto C, Hussaini MY, Quarteroni A, Zang TA (1987) Spectral methods in fluid dynamics. Springer, BerlinzbMATHGoogle Scholar
Cheng C, Liu Q, Lee J, Massoud HZ (2004) Spectral element method for the Schrödinger–Poisson system. J Comput Electron 3:417–421CrossRefGoogle Scholar
Castella F (1997) \(L^2\) solutions to the Schrödinger–Poisson system: existence, uniqueness, time behavior, and smoothing effects. Math Mod Meth Appl Sci 7:1051–1083CrossRefzbMATHGoogle Scholar
Dong X (2011) A short note on simplified pseudospectral methods for computing ground state and dynamics of spherically symmetric Schrödinger–Poisson-Slater system. J Comput Phys 230:7917–7922MathSciNetCrossRefzbMATHGoogle Scholar
Ehrhardt M, Zisowsky A (2006) Fast calculation of energy and mass preserving solutions of Schrödinger–Poisson systems on unbounded domains. J Comput Appl Math 187:1–28MathSciNetCrossRefzbMATHGoogle Scholar
Lubich C (2008) On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math Comput 77:2141–2153CrossRefzbMATHGoogle Scholar
Mauser NJ, Zhang Y (2014) Exact artificial boundary condition for the Poisson equation in the simulation of the 2D Schrödinger–Poisson system. Commun Comput Phys 16:764–780MathSciNetCrossRefzbMATHGoogle Scholar
Manfredi G, Haas F (2001) Self-consistent fluid model for a quantum electron gas. Phys Rev B 64:075316CrossRefGoogle Scholar