A finite difference method with meshless interpolation for incompressible flows in non-graded tree-based grids

Abstract Tree-based mesh grids bring the advantage of using fast cartesian discretizations, such as finite differences, and the flexibility and accuracy of local mesh refinement. The main challenge is how to adapt the discretization stencil near the interfaces between grid elements of different sizes, which is usually solved by local high-order geometrical interpolations. These interpolations depend on the distribution of cells in the vicinity of the point of interest, hence they are site-specific and can become quite complex in three-dimensional simulations, specially when dealing with staggered unknown arrangements. Most methods usually avoid this by limiting the mesh configuration (usually to graded quadtree/octree grids), reducing the number of cases to be treated locally. In this work, we propose a robust method based on a moving least squares meshless interpolation technique, which is employed to compute the weights of the finite difference approximation in a given hierarchical grid, allowing for complex mesh configurations, still keeping the overall order of accuracy of the resulting method. Numerical convergence tests and application to fluid flow simulations are performed to illustrate the flexibility, robustness and accuracy of this new approach.