High-order compact finite volume methods on unstructured grids with adaptive mesh refinement for solving inviscid and viscous flows

In the present paper, high-order finite volume schemes on unstructured grids developed in our previous papers are extended to solve three-dimensional inviscid and viscous flows. The high-order variational reconstruction technique in terms of compact stencil is improved to reduce local condition numbers. To further improve the efficiency of computation, the adaptive mesh refinement technique is implemented in the framework of high-order finite volume methods. Mesh refinement and coarsening criteria are chosen to be the indicators for certain flow structures. One important challenge of the adaptive mesh refinement technique on unstructured grids is the dynamic load balancing in parallel computation. To solve this problem, the open-source library p4est based on the forest of octrees is adopted. Several two- and three-dimensional test cases are computed to verify the accuracy and robustness of the proposed numerical schemes.