We present a second-order monolithic method for solving incompressible Navier--Stokes equations on irregular domains with quadtree grids. A semi-collocated grid layout is adopted, where velocity variables are located at cell vertices, and pressure variables are located at cell centers. Compact finite difference methods with ghost values are used to discretize the advection and diffusion terms of the velocity. A pressure gradient and divergence operator on the quadtree that use compact stencils are developed. Furthermore, the proposed method is extended to cubical domains with octree grids. Numerical results demonstrate that the method is second-order convergent in $L^\infty$ norms and can handle irregular domains for various Reynolds numbers.

翻译：我们提出了一种二级单极方法,用于解决有四叶网格的非常规域的不压缩纳维埃-斯托克方程式。采用了半对齐的网格布局,其速度变量位于细胞脊椎,压力变量位于细胞中心。使用带有幽灵价值的有限差异方法将速度的倾斜和扩散条件分解开来。开发了四叶的压力梯度和差异运算器,使用紧凑的斜线。此外,拟议方法还扩大到有八叶网格的立方域。数字结果显示,该方法以$ ⁇ infty$标准为第二级,可以处理各种 Reynolds数字的不规则域。