Energy-preserving exponential wave integrator method and the long-time dynamics for the two-dimensional space fractional coupled Klein–Gordon–Dirac equation

In this article, uniform error bounds of an exponential wave integrator Fourier pseudo-spectral (EWIFP) method are established for the long-time dynamics of two-dimensional nonlinear space fractional Klein–Gordon–Dirac equation (NSFKGDE) with a small coupling parameter $\varepsilon \in (0,1]$ and small potentials. By using the Fourier spectral discretization in space, followed with a second-order exponential wave integrator based on certain efficient quadrature rule in phase field, we construct a time-symmetric and energy-preserving numerical scheme. Rigorous analysis of uniform error bounds at $O\left(h^{m_0}+{\varepsilon }^{1-\gamma }{\tau }^2\right)$ for $\gamma \in [0,1]$ is carried out with the tool of cut-off technique as well as the energy method. Extensive numerical experiments demonstrate that the proposed discretization performs identically with our theoretical results. For applications, we profile the dynamical evolution of NSFKGDE in two dimensions (2D) with a honeycomb lattice potential, which correlates greatly with coupling, amplitude of potentials and fractional orders.