In this talk, an efficient and accurate numerical method will be proposed to compute the ground state of spin-2 Bose-Einstein condensates (BECs) by using the normalized gradient flow (NGF) or imaginary time method (ITM). The key idea is twofold. One is to find the five projection or normalization conditions that are used in the projection step of NGF/ITM, while the other one is to find a good initial data for the NGF/ITM. Based on the relations between chemical potentials and the two physical constrains given by the conservation of the totlal mass and magnetization, these five projection or normalization conditions can be completely and uniquely determined in the context of the the discrete scheme of the NGF discretized by back-Euler finite difference (BEFD) method, which allows one to successfully extend the most powerful and popular NGF/ITM to compute the ground state of spin-2 BECs. Additionally, the structures and properties of the ground states in a uniform system are analysed so as to construct efficient initial data for NGF/ITM. Extensive numerical results on ground states of spin-2 BECs with ferromagnetic/nematic/cyclic interaction and harmonic/optical lattice potential in one/two dimensions are reported to show the efficiency of our method and to demonstrate some interesting physical phenomena.

• Thematic program: Blow-up and Dispersion in nonlinear Schrödinger and Wave equations (2014)
• Event: Working group on "Efficient numerics for NLS" (2015)