In this talk, a normalized gradient flow (NGF) (or imaginary time method (ITM)) and a preconditioned conjugate gradient (PCG) scheme are proposed to compute the ground state of spin-2 Bose-Einstein condensates (BECs). The key idea is twofold. On one hand, the ground states in a spatial uniform system are analytically solved out and analyzed, which is helpful to analyze the  properties as well as simplify the computation of the ground states in spatial non-uniform system where single-mode-approximation is valid and useful to suggest initial data to efficiently initialize the NGF/ITM and PCG. On the other hand, three additional projection or normalization conditions based on the relations between chemical potentials should be found to determine the five projection parameters used in the projection step of the NGF and PCG. Hence, together with the two other physical constrains given by the  conservation of the total mass and magnetization, these five projection parameters can be determined. This allows one to successfully extend the  NGF/ITM  and PCG to compute the ground states of spin-2 BECs. Extensive numerical results on ground states of spin-2 BECs with ferromagnetic/nematic/cyclic phase and harmonic/optical lattice/box potential in one/two dimensions are reported to show the efficiency of our method and to demonstrate some interesting physical phenomena.

袁永军，男，博士，教授，湖南省青年芙蓉学者。2007年本科毕业于湖南师范大学大学数学与应用数学专业，2012年博士毕业于湖南师范大学计算数学专业， 2012年至2015年先后在北京计算科学研究中心和新加坡国立大学从事博士后研究工作，2015年11月进入湖南师范大学工作至今。主要从事非线性偏微分方程多解计算和玻色-爱因斯坦凝聚数值模拟方面研究，部分工作发表在Mathematics of Computation， SIAM Journal on Scientific Computing, Physical Review Letters, Journal of Computational Physics和Physical Review A等杂志。主持国家自然科学基金青年基金项目1项和面上项目2项。