Reviews of high-order methods on unstructured and hybrid grid

Abstract

While 2nd order methods are dominant in most compressible flow simulations, many types of problems, such as computational aeroacoustics (CAA), vortex-dominant flows and large eddy simulation (LES) of turbulent flows, call for higher order accuracy (third order and more). The main deficiency of widely available, second-order methods for the accurate simulations of the above-mentioned flows is the numerical diffusion and dissipation of vorticity to unacceptable level. Applications of high-order accu- rate, low-diffusion and low dissipation numerical methods can significantly alleviate this deficiency of the traditional second order methods, improve predictions of vortical and other complex, separated, unsteady flows. On the other hand, for complex configurations, the structured/unstructured hybrid grid technique presents the trend of grid generation technique. Therefore the high-order methods on unstructured and hybrid (or mixed) grids are paid much more attention in recent years. In this paper, the high-order methods on unstructured and hybrid grids are reviewed comprehensively, including the k-exact finite volume (FV) methods, the FV methods based on ENO and WENO reconstruction, the discontinuous Galerkin (DG) finite element method, the finite spectral volume (SV) methods, the finite spectral dif- ference (SD) method, DG/FV hybrid methods, the unified method based on correction procedure via reconstruction (CPR). The main ideas of these high-order methods are introduced, and the advantages and disadvantages of these methods are discussed. In addition, some important issues for simulations of complex geometry are discussed, including the treatment of curved boundary, the detector of discontinu-ity and high-order limiters, implicit iteration methods, and hp-multigrid approaches. We believe that the high-order methods on unstructured and hybrid grids will play more and more important role on more accurate simulations of realistic airspace vehicles and study of complex flow mechanism in the future.