High-order conservative schemes for the nonlinear Dirac equation

Abstract

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 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.