Minimal number of discrete velocities for a flow description and internal structural evolution of a shock wave

Chen, 1998, Lattice Boltzmann method for fluid flows, Annu. Rev. Fluid Mech., 30, 329, 10.1146/annurev.fluid.30.1.329 Chen, 2003, Extended Boltzmann kinetic equation for turbulent flows, Science, 301, 633, 10.1126/science.1085048 Rothman, 2004 Monaco, 1991 Shim, 2010, Robust thermal boundary conditions applicable to a wall along which temperature varies in lattice-gas cellular automata, Phys. Rev. E, 81, 10.1103/PhysRevE.81.046703 Philippi, 2006, From the continuous to the lattice Boltzmann equation: The discretization problem and thermal models, Phys. Rev. E, 73, 10.1103/PhysRevE.73.056702 Shan, 2010, General solution of lattices for Cartesian lattice Bhatanagar–Gross–Krook models, Phys. Rev. E, 81, 10.1103/PhysRevE.81.036702 Shim, 2013, Multidimensional on-lattice higher-order models in the thermal lattice Boltzmann theory, Phys. Rev. E, 88, 10.1103/PhysRevE.88.053310 Shim, 2013, How to obtain higher-order multivariate Hermite expansion of Maxwell–Boltzmann distribution by using Taylor expansion?, Z. Angew. Math. Phys., 64, 473, 10.1007/s00033-012-0265-1 Shim, 2017, Parametric lattice Boltzmann method, J. Comput. Phys., 338, 240, 10.1016/j.jcp.2017.02.057 Bellomo, 1991, The discrete Boltzmann equation: a review of the mathematical aspects of the initial and initial-boundary value problems, Rev. Math. Phys., 3, 137, 10.1142/S0129055X91000060 Frisch, 1991, Relation between the lattice Boltzmann equation and the Navier–Stokes equations, Physica D, 47, 231, 10.1016/0167-2789(91)90293-I Yang, 2017, Comparative study of discrete velocity method and high-order lattice Boltzmann method for simulation of rarefied flows, Comput. & Fluids, 146, 125, 10.1016/j.compfluid.2017.01.014 Su, 2017, Comparative study of the discrete velocity and lattice Boltzmann methods for rarefied gas flows through irregular channels, Phys. Rev. E, 96, 10.1103/PhysRevE.96.023309 Yoshino, 2004, Comparison of accuracy and efficiency between the lattice Boltzmann method and the finite difference method in viscous/thermal fluid flows, Int. J. Comput. Fluid Dyn., 18, 333, 10.1080/1061856031000154775 Grad, 1949, Note on N-dimensional Hermite polynomials, Comm. Pure Appl. Math., 2, 325, 10.1002/cpa.3160020402 Courant, 1999