A three-dimensional (3D) free-parameter multiple-relaxation-time lattice Boltzmann method for high speed compressible and low speed incompressible flows is presented. In the approach transformation matrix is constructed according to irreducible representation basis functions of SO(3) group. Equilibria of nonconserved moments are chosen so as to recover compressible Navier-Stokes equations through Chapman-Enskog analysis. Sizes of discrete velocities are flexible. Influence of model parameters on numerical stability is analyzed. Reference values of parameters are suggested. To validate performance of the model, several well-known benchmark problems ranging from 1D to 3D are simulated. Numerical results are in good agreement with analytical solutions and/or other numerical results.

The work discussed algorithm of grand canonical quantum Monte Carlo method for the one-band two-dimensional Hubbard model, and calculated local magnetic moment, magnetic susceptibility, staggered magnetic susceptibility and the internal energy of the model. Results show that grand canonical quantum Monte Carlo method is very effective in dealing with Hubbard model from weak to strong correlation rigim, and also in dealing with many——body problems with strong correlation.