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Improved uniform error bounds on time-splitting methods are rigorously provenfor the long-time dynamics of the weakly nonlinear Dirac equation (NLDE), wherethe nonlinearity strength is characterized by a dimensionless parameter$\varepsilon \in (0, 1]$ . We adopt a second order Strang splitting method todiscretize the NLDE in time and combine the Fourier pseudospectral method inspace for the full-discretization. By employing the {\sl regularitycompensation oscillation} (RCO) technique where the high frequency modes arecontrolled by the regularity of the exact solution and the low frequency modesare analyzed by phase cancellation and energy method, we establish improveduniform error bounds at $O(\varepsilon^2\tau^2)$ and $O(h^{m-1}+\varepsilon^2\tau^2)$ for the second-order Strang splitting semi-discretizaionand full-discretization up to the long-time $T_{\varepsilon} = T/\varepsilon^2$with $T>0$ fixed, respectively. Furthermore, the numerical scheme and errorestimates are extended to an oscillatory NLDE which propagates waves with$O(\varepsilon^2)$ wavelength in time. Finally, numerical examples verifyingour analytical results are given.

Improved uniform error bounds on time-splitting methods for the long-time dynamics of the weakly nonlinear Dirac equation