论文标题
改进的统一误差界定了弱非线性狄拉克方程的长时间动力学的时间分类方法
Improved uniform error bounds on time-splitting methods for the long-time dynamics of the weakly nonlinear Dirac equation
论文作者
论文摘要
对于弱非线性狄拉克方程（NLDE）的长期动态（NLDE）的长期动态，在时间序列方法上的改进均匀误差界限得到了严格的证明，其中非线性强度的特征是无尺寸参数$ \ varepsilon \ in（0，1] $中的无尺寸参数。全局部化。 \ varepsilon^2τ2）$用于二阶Strang分裂的半散制和全滴度，直至长期$ t _ {\ varepsilon} = t/\ varepsilon^2 $，$ t> $ t> $ t> 0 $。 $ o（\ varepsilon^2）$ peavellength的时间。
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 $O(\varepsilon^2τ^2)$ and $O(h^{m-1}+ \varepsilon^2τ^2)$ 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.
