Skip to main content
SpringerLink
Log in

A splitting Chebyshev collocation method for Schrödinger–Poisson system

  • Published:
Computational and Applied Mathematics Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  • Anderson D et al (2002) Statistical effects in the multistream model for quantum plasmas. Phys Rev E 65:046417

    Article  Google Scholar 

  • Becker KH, Schoenbach KH, Eden JG (2006) Microplasmas and applications. J Phys D 39:R55

    Article  Google 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 1:809–831

    Article  MATH  Google Scholar 

  • Abdallah N Ben (2000) On a multidimensional Schrödinger–Poisson scattering model for semiconductors. J Math Phys 41(7):4241–4261

    Article  MathSciNet  MATH  Google Scholar 

  • Brezzi F, Markowich PA (1991) The three-dimensional Wigner–Poisson problem: existence, uniqueness and approximation. Math Meth Appl Sci 14:35–61

    Article  MathSciNet  MATH  Google Scholar 

  • Canuto C, Hussaini MY, Quarteroni A, Zang TA (1987) Spectral methods in fluid dynamics. Springer, Berlin

    MATH  Google Scholar 

  • Cheng C, Liu Q, Lee J, Massoud HZ (2004) Spectral element method for the Schrödinger–Poisson system. J Comput Electron 3:417–421

    Article  Google Scholar 

  • Castella F (1997) \(L^2\) solutions to the Schrödinger–Poisson system: existence, uniqueness, time behavior, and smoothing effects. Math Mod Meth Appl Sci 7:1051–1083

    Article  MATH  Google 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 230:7917–7922

    Article  MathSciNet  MATH  Google 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 187:1–28

    Article  MathSciNet  MATH  Google Scholar 

  • Harrison R, Moroz IM, Tod KP (2003) A numerical study of Schrödinger–Newton equations. Nonlinearity 16:101–122

    Article  MathSciNet  MATH  Google Scholar 

  • Haas F, Manfredi G, Feix M (2000) Multistream model for quantum plasmas. Phys Rev E 62:2763

    Article  Google Scholar 

  • Haas F (2003) Quantum ion-acoustic waves. Phys Plasmas 10:3858–3866

    Article  Google Scholar 

  • Lange H, Toomire B, Zweifel PF (1995) An overview of Schrödinger–Poisson Problems. Rep Math Phys 36:331–345

    Article  MathSciNet  MATH  Google Scholar 

  • Lubich C (2008) On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math Comput 77:2141–2153

    Article  MATH  Google 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 16:764–780

    Article  MathSciNet  MATH  Google Scholar 

  • Manfredi G, Haas F (2001) Self-consistent fluid model for a quantum electron gas. Phys Rev B 64:075316

    Article  Google Scholar 

  • Manfredi G (2005) How to model quantum plasmas. Fields Inst Commun 46(263):2005

    MathSciNet  MATH  Google Scholar 

  • Markowich PA, Ringhofer CA, Schmeiser C (1990) Semiconductor equations. Springer, Berlin

    Book  MATH  Google Scholar 

  • Moroz I, Penrose R, Tod P (1998) Spherically-symmetric solutions of the Schrödinger–Newton equations. Class Quantum Grav 15:2733–2742

    Article  MATH  Google Scholar 

  • Opher M, Silva LO, Dauger DE, Decyk VK, Dawson JM (2001) Nuclear reaction rates and energy in stellar plasmas: the effect of highly damped modes. Phys Plasmas 8:2454–2460

    Article  Google Scholar 

  • Peyret R (2002) Spectral methods for incompressible viscous flow. Springer, New York

    Book  MATH  Google Scholar 

  • Shaikh D, Shukla PK (2008) 3D electron fluid turbulence at nanoscales in dense plasmas. New J Phys 10(083007):1–7

    Google Scholar 

  • Shen J (2006) Efficient spectral-Galerkin method II. direct solvers of second- and fourth-order equations using Chebyshev polynomials. SIAM J Sci Comput 16:74–87

    Article  MathSciNet  MATH  Google Scholar 

  • Shukla PK, Eliasson B (2010) Nonlinear aspects of quantum plasma physics. Phys Usp 53:51–76

    Article  Google Scholar 

  • Shukla PK, Stenflo L (2006) Stimulated scattering instabilities of electromagnetic waves in an ultracold quantum plasma. Phys Plasmas 13:044505

    Article  Google Scholar 

  • Shukla PK, Eliasson B (2007) Nonlinear interactions between electromagnetic waves and electron plasma oscillations in quantum plasmas. Phys Rev Lett 99:096401

    Article  Google Scholar 

  • Shukla PK, Eliasson B (2006) Formation and dynamics of dark solitons and vortices in quantum electron plasmas. Phys Rev Lett 96:245001

    Article  Google Scholar 

  • Sulem C, Sulem PL (1999) The nonlinear Schrödinger equation: self-focusing and wave collapse. Springer, Berlin

    MATH  Google Scholar 

  • Tan IH, Snider GL, Chang LD, Hu EL (1990) A self-consistent solution of Schrödinger–Poisson equations using a nonuniform mesh. J Appl Phys 68:4071–4076

    Article  Google Scholar 

  • Tod P, Moroz IM (1999) An analytical approach to the Schrödinger–Newton equations. Nonlinearity 12:201–216

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang Y (2013) Optimal error estimates of compact finite difference discretizations for the Schrödinger–Poisson system. Commun Comput Phys 13:1357–1388

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang Y, Dong XC (2011) On the computation of ground state and dynamics of Schrödinger–Poisson–Slater system. J Comput Phys 230:2660–2676

    Article  MathSciNet  MATH  Google Scholar 

  • Zheng C (2007) A perfectly matched layer approach to the nonlinear Schrödinger wave equations. J Comput Phys 227:537–556

    Article  MathSciNet  MATH  Google Scholar 

  • Wang Y, Lu X (2014) Modulational instability of electrostatic acoustic waves in an electron-hole semiconductor quantum plasma. Phys Plasma 21:022107

    Article  Google Scholar 

  • Wang H (2010) An efficient Chebyshev–Tau spectral method for Ginzburg–Landau–Schrödinger equations. Comput Phys Commun 181:325–340

    Article  MATH  Google Scholar 

Download references

Acknowledgements

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.

Author information

Authors and Affiliations

  1. School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, 650221, Yunnan, People’s Republic of China

    Hanquan Wang & Ronghua Liu

  2. School of Mathematical Sciences, Fudan University, 220 Handan Road, Shanghai, 200433, People’s Republic of China

    Zhenguo Liang

Authors
  1. Hanquan Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhenguo Liang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ronghua Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hanquan Wang.

Additional information

Communicated by Pierangelo Marcati.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, H., Liang, Z. & Liu, R. A splitting Chebyshev collocation method for Schrödinger–Poisson system. Comp. Appl. Math. 37, 5034–5057 (2018). https://doi.org/10.1007/s40314-018-0616-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40314-018-0616-4

Keywords

Mathematics Subject Classification

Search

Navigation