Solving incompressible Navier--Stokes equations on irregular domains and quadtrees by monolithic approach

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.