High-order methods for the Euler and Navier-Stokes equations on unstructured grids

This article reviews several unstructured grid-based high-order methods for the compressible Euler and Navier-Stokes equations. We treat the spatial and temporal discretizations separately, hoping that it is easier to spot the similarities and differences of various types of methods. Our main focus is to present the basic design principles of each method, and highlight its pros and cons when appropriate. Sample computational results are shown to illustrate the capability of selected methods. These high-order methods are expected to be more efficient than low-order methods for problems requiring high accuracy, such as wave propagation problems, vortex-dominated flows including high-lift configuration, helicopter blade vortex interaction, as well as large eddy simulation and direct numerical simulation of turbulence. We conclude the paper with several current challenges in the proliferation of high-order methods in the aerospace community.