报告题目：Super-resolution property of splitting methods for Dirac equation in the nonrelativistic limit regime
报告地点：尚贤楼706报告厅
报告时间：2019年4月17日（星期三）上午10:00--11:00
主持人：王廷春副教授
报告人简介：蔡勇勇博士，国家青年千 人计划入选者，本科和硕士就读于北京大学数学科学学院，2012年于新加坡国立大学数学系获博士学位，后至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，2016年5月起，任北京计算科学研究中心特聘研究员。目前主持一项国家自然科学基金面上项目，参与承担一项国家自然科学基金重点项目。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应用，相关研究结果发表在SIAM Journal on Numerical Analysis、SIAM Journal on Applied Mathematics、SIAM Journal on Mathematical Analysis、Mathematics of Computation、Journal of Computational Physics、Journal of Scientific Computing、Journal of Functional Analysis等学术期刊上。
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报告摘要：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. Finally, numerical examples are reported to validate our findings.
数学与统计学院
2019年4月12日