Super-resolution 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 super-resolution, 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 non-resonant, i.e. $\tau$ is away from 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 super-resolution is still valid for higher order splitting methods.
翻译:利塔断裂的超级分辨率(S_1美元)和斯特朗分解的超级分辨率(S_2美元)在非线性迪拉克方程式中受到严格分析,在非相对性制度中,无外部磁潜能的非线性迪拉克方程式,其参数小,为0瓦列普西隆/leq 1美元,与光速成反比。在这个制度中,与波长(O美元)和(瓦列普西隆2美元)相比,溶解的高度振动。分解方法令人惊讶地显示超解,即,即使时间步数(美元)大于(瓦列普西隆2美元)的抽样波长,该方法也能够准确找到解决办法。与线性案件类似,1美元和2美元,两者均显示1美元一致。此外,如果美元是非共值,即美元与美元相比,即,美元比由美元固定值确定的某些区域准确度(美元)高于美元。 美元-S/2美元将使得统一的结果得到改进。