A High-Order Accurate Gas-Kinetic Scheme for One- and Two-Dimensional Flow Simulation (Communications in Computational Physics, 2020)
This paper develops a high-order accurate gas-kinetic scheme in the frameworkof the finite volume method for the one- and two-dimensional flow simulations,which is an extension of the third-order accurate gas-kinetic scheme [Q.B. Li, K. Xu, andS. Fu, J. Comput. Phys., 229(2010), 6715-6731] and the second-order accurate gas-kineticscheme [K. Xu, J. Comput. Phys., 171(2001), 289-335]. It is formed by two parts: quarticpolynomial reconstruction of the macroscopic variables and fourth-order accurate fluxevolution. The first part reconstructs a piecewise cell-center based quartic polynomialand a cell-vertex based quartic polynomial according to the "initial" cell average approximationof macroscopic variables to recover locally the non-equilibrium and equilibriumsingle particle velocity distribution functions around the cell interface. It is inview of the fact that all macroscopic variables become moments of a single particle velocitydistribution function in the gas-kinetic theory. The generalized moment limiteris employed there to suppress the possible numerical oscillation. In the second part,the macroscopic flux at the cell interface is evolved in fourth-order accuracy by meansof the simple particle transport mechanism in the microscopic level, i.e. free transportand the Bhatnagar-Gross-Krook (BGK) collisions. In other words, the fourth-orderflux evolution is based on the solution (i.e. the particle velocity distribution function)of the BGK model for the Boltzmann equation. Several 1D and 2D test problems arenumerically solved by using the proposed high-order accurate gas-kinetic scheme. Bycomparing with the exact solutions or the numerical solutions obtained the second-orderor third-order accurate gas-kinetic scheme, the computations demonstrate thatour scheme is effective and accurate for simulating invisid and viscous fluid flows,and the accuracy of the high-order GKS depends on the choice of the (numerical) collisiontime.