We propose a novel collocated projection method for solving the incompressible Navier-Stokes equations with arbitrary boundaries. Our approach employs non-graded octree grids, where all variables are stored at the nodes. To discretize the viscosity and projection steps, we utilize supra-convergent finite difference approximations with sharp boundary treatments. We demonstrate the stability of our projection on uniform grids, identify a sufficient stability condition on adaptive grids, and validate these findings numerically. We further demonstrate the accuracy and capabilities of our solver with several canonical two- and three-dimensional simulations of incompressible fluid flows. Overall, our method is second-order accurate, allows for dynamic grid adaptivity with arbitrary geometries, and reduces the overhead in code development through data collocation.

翻译：我们提出了一种新颖的共置投影方法，用于求解具有任意边界的不可压缩Navier-Stokes方程。该方法采用非均匀八叉树网格，所有变量均存储于节点之上。为离散粘性项与投影步骤，我们利用具有尖锐边界处理的超收敛有限差分逼近。我们在均匀网格上证明了该投影方法的稳定性，识别了自适应网格上的充分稳定性条件，并通过数值验证了这些结论。此外，我们通过多个典型二维与三维不可压缩流体流动的模拟，进一步验证了求解器的精度与能力。总体而言，该方法具有二阶精度，支持任意几何形状的动态网格自适应，并通过数据共置降低了代码开发的开销。