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

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