A splitting Chebyshev collocation method for Schrödinger–Poisson system

Article
  • 89 Downloads

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.

Keywords

Nonlinear Schrödinger and Poisson system Chebyshev collocation method Splitting method Quantum plasmas 

Mathematics Subject Classification

35Q55 65Z05 65N12 65N35 

Notes

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.

References

  1. Anderson D et al (2002) Statistical effects in the multistream model for quantum plasmas. Phys Rev E 65:046417CrossRefGoogle Scholar
  2. Becker KH, Schoenbach KH, Eden JG (2006) Microplasmas and applications. J Phys D 39:R55CrossRefGoogle Scholar
  3. 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 1:809–831CrossRefMATHGoogle Scholar
  4. Abdallah N Ben (2000) On a multidimensional Schrödinger–Poisson scattering model for semiconductors. J Math Phys 41(7):4241–4261MathSciNetCrossRefMATHGoogle Scholar
  5. Brezzi F, Markowich PA (1991) The three-dimensional Wigner–Poisson problem: existence, uniqueness and approximation. Math Meth Appl Sci 14:35–61MathSciNetCrossRefMATHGoogle Scholar
  6. Canuto C, Hussaini MY, Quarteroni A, Zang TA (1987) Spectral methods in fluid dynamics. Springer, BerlinMATHGoogle Scholar
  7. Cheng C, Liu Q, Lee J, Massoud HZ (2004) Spectral element method for the Schrödinger–Poisson system. J Comput Electron 3:417–421CrossRefGoogle Scholar
  8. Castella F (1997) \(L^2\) solutions to the Schrödinger–Poisson system: existence, uniqueness, time behavior, and smoothing effects. Math Mod Meth Appl Sci 7:1051–1083CrossRefMATHGoogle Scholar
  9. 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 230:7917–7922MathSciNetCrossRefMATHGoogle Scholar
  10. Ehrhardt M, Zisowsky A (2006) Fast calculation of energy and mass preserving solutions of Schrödinger–Poisson systems on unbounded domains. J Comput Appl Math 187:1–28MathSciNetCrossRefMATHGoogle Scholar
  11. Harrison R, Moroz IM, Tod KP (2003) A numerical study of Schrödinger–Newton equations. Nonlinearity 16:101–122MathSciNetCrossRefMATHGoogle Scholar
  12. Haas F, Manfredi G, Feix M (2000) Multistream model for quantum plasmas. Phys Rev E 62:2763CrossRefGoogle Scholar
  13. Haas F (2003) Quantum ion-acoustic waves. Phys Plasmas 10:3858–3866CrossRefGoogle Scholar
  14. Lange H, Toomire B, Zweifel PF (1995) An overview of Schrödinger–Poisson Problems. Rep Math Phys 36:331–345MathSciNetCrossRefMATHGoogle Scholar
  15. Lubich C (2008) On splitting methods for Schrödinger–Poisson and cubic nonlinear Schrödinger equations. Math Comput 77:2141–2153CrossRefMATHGoogle Scholar
  16. 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 16:764–780MathSciNetCrossRefMATHGoogle Scholar
  17. Manfredi G, Haas F (2001) Self-consistent fluid model for a quantum electron gas. Phys Rev B 64:075316CrossRefGoogle Scholar
  18. Manfredi G (2005) How to model quantum plasmas. Fields Inst Commun 46(263):2005MathSciNetMATHGoogle Scholar
  19. Markowich PA, Ringhofer CA, Schmeiser C (1990) Semiconductor equations. Springer, BerlinCrossRefMATHGoogle Scholar
  20. Moroz I, Penrose R, Tod P (1998) Spherically-symmetric solutions of the Schrödinger–Newton equations. Class Quantum Grav 15:2733–2742CrossRefMATHGoogle Scholar
  21. 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 8:2454–2460CrossRefGoogle Scholar
  22. Peyret R (2002) Spectral methods for incompressible viscous flow. Springer, New YorkCrossRefMATHGoogle Scholar
  23. Shaikh D, Shukla PK (2008) 3D electron fluid turbulence at nanoscales in dense plasmas. New J Phys 10(083007):1–7Google Scholar
  24. Shen J (2006) Efficient spectral-Galerkin method II. direct solvers of second- and fourth-order equations using Chebyshev polynomials. SIAM J Sci Comput 16:74–87MathSciNetCrossRefMATHGoogle Scholar
  25. Shukla PK, Eliasson B (2010) Nonlinear aspects of quantum plasma physics. Phys Usp 53:51–76CrossRefGoogle Scholar
  26. Shukla PK, Stenflo L (2006) Stimulated scattering instabilities of electromagnetic waves in an ultracold quantum plasma. Phys Plasmas 13:044505CrossRefGoogle Scholar
  27. Shukla PK, Eliasson B (2007) Nonlinear interactions between electromagnetic waves and electron plasma oscillations in quantum plasmas. Phys Rev Lett 99:096401CrossRefGoogle Scholar
  28. Shukla PK, Eliasson B (2006) Formation and dynamics of dark solitons and vortices in quantum electron plasmas. Phys Rev Lett 96:245001CrossRefGoogle Scholar
  29. Sulem C, Sulem PL (1999) The nonlinear Schrödinger equation: self-focusing and wave collapse. Springer, BerlinMATHGoogle Scholar
  30. Tan IH, Snider GL, Chang LD, Hu EL (1990) A self-consistent solution of Schrödinger–Poisson equations using a nonuniform mesh. J Appl Phys 68:4071–4076CrossRefGoogle Scholar
  31. Tod P, Moroz IM (1999) An analytical approach to the Schrödinger–Newton equations. Nonlinearity 12:201–216MathSciNetCrossRefMATHGoogle Scholar
  32. Zhang Y (2013) Optimal error estimates of compact finite difference discretizations for the Schrödinger–Poisson system. Commun Comput Phys 13:1357–1388MathSciNetCrossRefMATHGoogle Scholar
  33. Zhang Y, Dong XC (2011) On the computation of ground state and dynamics of Schrödinger–Poisson–Slater system. J Comput Phys 230:2660–2676MathSciNetCrossRefMATHGoogle Scholar
  34. Zheng C (2007) A perfectly matched layer approach to the nonlinear Schrödinger wave equations. J Comput Phys 227:537–556MathSciNetCrossRefMATHGoogle Scholar
  35. Wang Y, Lu X (2014) Modulational instability of electrostatic acoustic waves in an electron-hole semiconductor quantum plasma. Phys Plasma 21:022107CrossRefGoogle Scholar
  36. Wang H (2010) An efficient Chebyshev–Tau spectral method for Ginzburg–Landau–Schrödinger equations. Comput Phys Commun 181:325–340CrossRefMATHGoogle Scholar

Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2018

Authors and Affiliations

  1. 1.School of Statistics and MathematicsYunnan University of Finance and EconomicsKunmingPeople’s Republic of China
  2. 2.School of Mathematical SciencesFudan UniversityShanghaiPeople’s Republic of China

Personalised recommendations