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Lattice Boltzmann model for the one-dimensional nonlinear Dirac equation.

Baochang Shi1, Zhaoli Guo

  • 1School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, People's Republic of China. shibc@hust.edu.cn

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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A new lattice Boltzmann model accurately simulates the one-dimensional nonlinear Dirac equation. This numerical method demonstrates second-order accuracy in space and time, proving effective for complex physics problems.

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Area of Science:

  • Computational Physics
  • Numerical Analysis
  • Quantum Mechanics

Background:

  • The nonlinear Dirac equation describes various phenomena in quantum field theory and condensed matter physics.
  • Accurate numerical methods are crucial for solving these complex equations.
  • Traditional methods can be computationally intensive or limited in applicability.

Purpose of the Study:

  • To present a novel lattice Boltzmann model for the one-dimensional nonlinear Dirac equation.
  • To investigate the numerical accuracy and stability of the proposed model.
  • To assess the model's effectiveness for simulating nonlinear Dirac phenomena.

Main Methods:

  • Development of a lattice Boltzmann model using double complex-valued distribution functions.
  • Careful selection of equilibrium distribution functions tailored for the nonlinear Dirac equation.
  • Numerical investigation of the model's performance concerning space/time resolutions and relaxation time.

Main Results:

  • The lattice Boltzmann model achieves second-order accuracy in both spatial and temporal dimensions.
  • Near third-order accuracy was observed at lower grid resolutions, indicating robust performance.
  • The model demonstrated stability under various numerical conditions.

Conclusions:

  • The proposed lattice Boltzmann method is an effective and accurate numerical scheme for the one-dimensional nonlinear Dirac equation.
  • The model's performance validates the lattice Boltzmann approach for complex nonlinear wave phenomena.
  • This work provides a valuable tool for researchers in fields utilizing the nonlinear Dirac equation.