Abstract: 

We establish error bounds of the Lie-Trotter splitting and Strang splitting for the Dirac equation in the nonrelativistic limit regime in the absence of external magnetic potentials. In this regime, the solution admits high frequency waves in time. Surprisingly, we find out that the splitting methods exhibit super-resolutions, i.e. the methods can capture the solutions accurately even if the time step size is much larger than the sampled wavelength. Lie splitting shows half order uniform convergence w.r.t temporal wave length. Moreover, if the time step size is non-resonant, Lie splitting would yield an improved uniform first order uniform error bound. In addition, we show Strang splitting is uniformly convergent with half order rate for general time step size and uniformly convergent with three half order rate for non-resonant time step size. We also discuss the case with external magnetic potentials, and splitting schemes also show superior performance among the commonly used numerical methods.

个人简介：

蔡勇勇，北京师范大学教授，本科和硕士就读于北京大学，2012年在新加坡国立大学获得博士学位。他先后在威斯康辛大学麦迪逊分校、马里兰大学帕克分校和普渡大学从事博士后研究工作，从2016年至2019年在北京计算科学研究中心任特聘研究员。蔡勇勇博士的研究兴趣主要是偏微分方程的数值方法及其在量子力学等领域中的应用。