SIAM Journal on Numerical Analysis, Volume 59, Issue 2, Page 1040-1066, January 2021.
Superresolution of the Lie--Trotter splitting ($S_1$) and Strang splitting ($S_2$) is rigorously analyzed for the nonlinear Dirac equation without external magnetic potentials in the nonrelativistic regime with a small parameter $0<{\varepsilon}\leq 1$ inversely proportional to the speed of light. In this regime, the solution highly oscillates in time with wavelength at $O({\varepsilon}^2)$. The splitting methods surprisingly show superresolution, i.e., the methods can capture the solution accurately even if the time step size $\tau$ is much larger than the sampled wavelength at $O({\varepsilon}^2)$. Similar to the linear case, $S_1$ and $S_2$ both exhibit $1/2$ order convergence uniformly with respect to ${\varepsilon}$. Moreover, if $\tau$ is nonresonant, i.e., $\tau$ is away from a certain region determined by ${\varepsilon}$, $S_1$ would yield an improved uniform first order $O(\tau)$ error bound, while $S_2$ would give improved uniform $3/2$ order convergence. Numerical results are reported to confirm these rigorous results. Furthermore, we note that superresolution is still valid for higher order splitting methods.

中文翻译：

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非相对论条件下无磁势的非线性狄拉克方程时间分解方法的一致误差界

SIAM数值分析杂志，第59卷，第2期，第1040-1066页，2021年1月。
严格分析了非相对论状态下具有较小参数$ 0 <{\ varepsilon} \ leq 1的非线性Dirac方程的Lie-Trotter分裂（$ S_1 $）和Strang分裂（$ S_2 $）的超分辨率$与光速成反比。在这种情况下，解决方案会随着时间以$ O（{\ varepsilon} ^ 2）$的波长高度振荡。分裂方法令人惊讶地显示出超分辨率，即，即使时间步长$ \ tau $远大于在$ O（{\ varepsilon} ^ 2）$处的采样波长，该方法也可以准确地捕获解。与线性情况类似，$ S_1 $和$ S_2 $相对于$ {\ varepsilon} $均表现出$ 1/2 $阶收敛。此外，如果$ \ tau $是非谐振的，即$ \ tau $远离由$ {\ varepsilon} $确定的某个区域，$ S_1 $将产生改进的统一一阶$ O（tau）$错误界限，而$ S_2 $将提供改进的统一$ 3/2 $阶收敛。报告了数值结果，以证实这些严格的结果。此外，我们注意到，超分辨率对于高阶拆分方法仍然有效。