Abstract: Over the past two decades there have been many research activities in the design and application of high order accurate numerical methods in computational fluid dynamics (CFD). High order methods are especially desirable for simulating flows with complicated solution structures. We give a review on the development and application of several classes of high order schemes in CFD, mainly concentrating on the simulation of compressible flows. An important feature of the compressible flow is the existence of shocks, interfaces and other discontinuities and often also complicated structure in the smooth part of the solution. This gives a unique challenge to the design of high order schemes to be non-oscillatory and yet still maintaining their high order accuracy. We concentrate our discussion on the essentially non-oscillatory (ENO), weighted ENO (WENO) finite difference,finite volume schemes and discontinuous Galerkin (DG) finite element methods. We attempt to describe their main properties and their relative strength and weakness. We also briefly review their developments and applications, concentrating mainly on the results over the past five years.

Key words: essentially non-oscillatory(ENO), weighted essentially non-oscillatory(WENO), discontinuous Galerkin(DG), high order accuracy, finite difference, finite volume, finite element, computational fluid dynamics, compressible flow