High-order conservative schemes for the nonlinear Dirac equation

ABSTRACT

The fine physical details for the quantum computation can easily be provided by high-order conservative schemes. By compensating the high-order difference operator $(D_{\mu p}-1)\delta$ to the central difference operator δ, a time-dependent semi-discrete system, which conserves both charge and energy, is derived for the two-dimensional nonlinear Dirac equation (NLDE). Two kinds of fully discrete schemes are obtained by discretizing this semi-discrete system in time with the time-midpoint method and the time-splitting method. We prove theoretically that the former one conserves both charge and energy while the latter one only keeps the charge conservation. Some numerical experiments are given to verify the accuracy order, the stability and the conservative properties. In addition, the dynamics of the NLDE in one dimension and two dimensions are simulated.

Keywords:

• Nonlinear Dirac equation
• operator-compensation
• high-order conservative scheme
• dynamics

2010 AMS subject classifications:

• 35L05
• 81-08
• 65M06

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (Project No. 11671044), the Science Challenge Project (Project No. TZ2016001), Foundation of Laboratory of Computational Physics (Project No. 6142A05180103) and Hunan Provincial Innovation Foundation for Postgraduate (Project No. CX2018B380).