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

TL;DR

This paper rigorously proves improved uniform error bounds for second-order time-splitting and spectral methods applied to the long-time dynamics of the weakly nonlinear Dirac equation, including oscillatory cases.

Contribution

It introduces the RCO technique to establish sharper error bounds for time-splitting methods on the NLDE over long times, extending to oscillatory regimes.

Findings

Error bounds of O(ε²τ²) for semi-discretization

Error bounds of O(h^{m-1}+ ε²τ²) for full discretization

Numerical results confirm the theoretical error estimates

Abstract

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