Convergence Analysis Of Finite Difference Fourier Pseudo-spectral Methods For The Nonlinear Dirac Equation

In this paper,we propose two finite-difference Fourier pseudo-spectral methods for the nonlinear Dirac equation with periodic boundary conditions.The general implementation of these two methods are as follows:the equation is discretized by the Fourier pseudo-spectral method in space and approximated by the finite difference method in time,the numerical schemes established by two methods are implicit,time symmetric and preserve the discrete mass and energy.The schemes are second-order accuracy in time and spectral accuracy in space without any restrictions on the grid ratio under the Hmnorm.The energy method,the cut-off technique of the nonlinearity and the mathematical induction method are used in the process of convergence analysis.Furthermore,the two numerical methods we proposed are still applicable to the two dimensional nonlinear Dirac equation and the nonlinear Dirac equation in the nonrelativistic limit regime.The solution is highly oscillatory in time in the nonrelativistic limit regime,so the ε-scalability of these two numerical methods is τ=O(ε3)and h=O(1),where h is mesh size andτ is time step.Finally,a large number of numerical results are showed to support our theoretical analysis,and we have also simulated the relevant dynamics of the nonlinear Dirac equation,such as honeycomb lattice potential and the collision of multiple solitary waves and so on.