Quadrature with equilibrium offset and its application on adaptive velocity grid

Due to the presence of the steep slope and concentration of the gas distribution function in the velocity space, local velocity grid adaptation becomes necessary for high speed aerospace applications and/or other high Knudsen flow. However, the adaptation of velocity grid complicates the quadrature of discrete distribution function, degrades the accuracy even with a large number of velocity grid points. This study presents a simple technique, termed as equilibrium offset, to modify quadrature in the velocity space, which separates the quadrature into an analytical integral of equilibrium distribution and a numerical quadrature of the derivation from the equilibrium. Modified quadrature provides accurate moments of the reference equilibrium state. As the numbers of velocity points increases, the modified quadrature converges to the real value of the integral regardless of the reference equilibrium state. We test several kinds of quadrature on nonuniform velocity grid. The results show that the proposed modification is superior as the distribution function is close to the equilibrium state. For the highly non-equilibrium distribution function the modified quadrature is equally good as the original quadrature. The equilibrium offset is implemented on the adaptive velocity grid. Several numerical simulations demonstrate the efficiency of the modified quadrature on adaptive velocity grid.