(untitled) (Communications in Computational Physics, 2025)
Cartesian grid method and local grid refinement for discrete velocity method (DVM) are developed in this work, with the numerical flux of DVM constructed by semi-Implicit Richtmyer method. To implement the boundary condition, the interpolation approach is applied, where the distribution function at the fluid side of the boundary point is first approximated by interpolation method given the knowledge of the boundary point. Once the distribution function is interpolated, the reflected Maxwellian distribution and numerical flux at boundary point can be evaluated. Then the distribution function at the fluid point close to the boundary can be updated by finite difference formulation. However, the interpolation of the distribution function at the boundary point is a significant challenge as the breakdown of the upwind stencil will cause instability. To preserve the upwind stencil, the most effective approach is to perform interpolation along the characteristic lines. Moreover, the local grid refinement is introduced to reduce the computational cost for industrial application. To validate the proposed Cartesian grid method, some numerical examples are simulated. The results demonstrate the accuracy and stability of the present method for straight boundary with oblique and curved boundary, subsonic and supersonic flows.