The Study On Exponential Wave Integrator Fourier Pseudo–spectral Method With Structure-preservation For The Dirac Equation With Small Electromagnetic Potentials

For the Dirac equation with small electromagnetic potentials characterized by a small parameter ε∈(0,1],the numerical methods for long-time dynamics have received more and more attention.Recently,two exponential wave integrator Fourier pseudo-spectral methods for the Dirac equation with small electromagnetic potentials have been proposed which are uniformly accurate about ε and perform well over the classical methods.However,these two methods can not preserve the mass and energy,which are important structural features of the Dirac equation from the perspective of geometric numerical integration.In addition,the methods are not time symmetric or only are conditionally stable under specific stability condition which implies CFL condition restrictions on the grid ratio.In this work,we propose a structure-preserving exponential wave integrator Fourier pseudo-spectral method.The proposed method is proved to be time symmetric,stable only under the condition τ（?）1 and preserves the discrete energy and modified mass.Without any CFL condition restrictions on the grid ratio,we carry out a rigourously error analysis and give uniform error bound of the method at O(h^mo+ε¹-β_τ²)up to the time at O（1/ε^β）with β∈[0,1],mesh size h,time step τ and an integer m₀ determined by the regularity conditions.In this paper,the numerical method is extended to a two-dimensional Dirac equation for analysis and numerical verification.In general,the Dirac equation with small electromagnetic potentials can be converted to an oscillatory Dirac equation with wavelength at O（ε^β）in time which includes the case of simultaneous massless and nonrelativistic regime.It is easy to extend the error bound and structurepreservation properties to the oscillatory Dirac equation.In addition,we extend this method to the weak nonlinear Dirac equation.Numerical experiments support our error bounds and structure-preservation properties.