scholarly journals Uniform Error Bounds of Time-Splitting Methods for the Nonlinear Dirac Equation in the Nonrelativistic Regime without Magnetic Potential

2021 ◽  
Vol 59 (2) ◽  
pp. 1040-1066
Author(s):  
Weizhu Bao ◽  
Yongyong Cai ◽  
Jia Yin
Keyword(s):  
Dirac Equation ◽  
Error Bounds ◽  
Splitting Methods ◽  
Magnetic Potential ◽  
Uniform Error ◽  
Nonlinear Dirac Equation ◽  
Time Splitting

A uniformly accurate (UA) multiscale time integrator pseudospectral method for the nonlinear Dirac equation in the nonrelativistic limit regime

2018 ◽  
Vol 52 (2) ◽  
pp. 543-566 ◽  
Author(s):  
Yongyong Cai ◽  
Yan Wang
Keyword(s):  
Dirac Equation ◽  
Convergence Rate ◽  
Error Estimates ◽  
Error Bounds ◽  
Linear Convergence ◽  
Mesh Size ◽  
Time Step ◽  
Nonlinear Dirac Equation ◽  
Temporal Oscillation ◽  
Exponential Wave Integrator

A multiscale time integrator Fourier pseudospectral (MTI-FP) method is proposed and rigorously analyzed for the nonlinear Dirac equation (NLDE), which involves a dimensionless parameter ε ∈ (0, 1] inversely proportional to the speed of light. The solution to the NLDE propagates waves with wavelength O (ε2) and O (1) in time and space, respectively. In the nonrelativistic regime,i.e., 0 < ε ≪ 1, the rapid temporal oscillation causes significantly numerical burdens, making it quite challenging for designing and analyzing numerical methods with uniform error bounds inε ∈ (0, 1]. The key idea for designing the MTI-FP method is based on adopting a proper multiscale decomposition of the solution to the NLDE and applying the exponential wave integrator with appropriate numerical quadratures. Two independent error estimates are established for the proposed MTI-FP method as hm0+τ2/ε2andhm0 + τ2 + ε2, where his the mesh size, τis the time step and m0depends on the regularity of the solution. These two error bounds immediately suggest that the MTI-FP method converges uniformly and optimally in space with exponential convergence rate if the solution is smooth, and uniformly in time with linear convergence rate at O (τ) for all ε ∈ (0, 1] and optimally with quadratic convergence rate at O (τ2) in the regimes when either ε = O (1) or 0 < ε ≲ τ. Numerical results are reported to demonstrate that our error estimates are optimal and sharp.

Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation

2014 ◽  
Vol 6 (4) ◽  
pp. 494-514 ◽  
Author(s):  
Yaming Chen ◽  
Songhe Song ◽  
Huajun Zhu
Keyword(s):  
Dirac Equation ◽  
Hamiltonian System ◽  
Numerical Experiments ◽  
Pseudospectral Method ◽  
Splitting Methods ◽  
Fourier Pseudospectral Method ◽  
Composition Method ◽  
Nonlinear Dirac Equation ◽  
Symplectic Scheme ◽  
Symplectic System

AbstractIn this paper, we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac (NLD) equation. Based on its multi-symplectic formulation, the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system. Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem, respectively. And the nonlinear subsystem is solved by a symplectic scheme. Finally, a composition method is applied to obtain the final schemes for the NLD equation. We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly. Numerical experiments are presented to show the effectiveness of the proposed methods.

scholarly journals On the nonlinear Dirac equation on noncompact metric graphs

2021 ◽  
Vol 278 ◽  
pp. 326-357
Author(s):  
William Borrelli ◽  
Raffaele Carlone ◽  
Lorenzo Tentarelli
Keyword(s):  
Dirac Equation ◽  
Metric Graphs ◽  
Nonlinear Dirac Equation

scholarly journals Solitary waves in the nonlinear Dirac equation in the presence of external driving forces

2016 ◽  
Vol 49 (6) ◽  
pp. 065402 ◽  
Author(s):  
Franz G Mertens ◽  
Fred Cooper ◽  
Niurka R Quintero ◽  
Sihong Shao ◽  
Avinash Khare ◽  
...  
Keyword(s):  
Dirac Equation ◽  
Solitary Waves ◽  
Driving Forces ◽  
Nonlinear Dirac Equation ◽  
External Driving
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