What's the future use for highorder END scheme
Hi, everyone
Now, I'm working on highorder ENO schemes for complex flow simulations. I'd like to get more wide commentary for the future use of highorder ENO schemes mainly based on Harton's method, as well as compareson with highorder compact schemes. Thanks in advance Yang 
Re: What's the future use for highorder END schem
ENO and WENO are originally designed for shockcapturing. The basic idea behind these two methods is that an upwind scheme will be dominant near the discontinuity, which definitely will lead to a distinct numerical dissipation part in the scheme. For general application in laminar flow, ENO or WENO is not a good choice because you have to spend much time in selecing the grid stencil. In these cases, highorder compact scheme may perform better. For turbulent flow, there is no general idea if this dissipation in ENO or WENO will damp the turbulent fluctuations. Nevertheless, compact scheme has been successfully used in turbulence simulation. With the increasing of the formal order of ENO and WENO( they can be 911th order, see a JCP paper in 2000), the numerical dissipation inside the scheme tend to be smaller, the main problem in using ENO and WENO thus is their CPU intensivity. In my conclusion 1) For highresolution shockcapturing, ENO or WENO is
highly recommanded. 2) For laminar flow, other highorder method is preferred unless you don't care the computation time and want to use the dissipation in the ENO/WENO to stabilize your computation. 3) For turbulent flow, you had better use other schemes. 
Re: What's the future use for highorder END schem
Hi, Paul Thanks for your response. I basically agree with your argument. As you know, ENO scheme is based on piecewise polynomial reconstruction, and this reconstruction can be extended to 2D and 3D problem with the full base of polynomial terms, that to say, including all highorder cross terms. But, highorder compact schemes are derived from one direction stencil, so, for 2D or 3D cases, highorder cross terms were neglected. I'm wondering which kind of effects these cross terms will take for the simulation of turbulent flow or some other complex flow. If these cross terms are neglected from ENO scheme, the CPU time will drop quite a lot. By the way, my another question is: what's the relation between highorder discretization scheme and highorder grid transformation?

Re: What's the future use for highorder END schem
Hello, Yang. ENO/WENO do use the method your described in highdimensionextension. And they also use conventional method (one direction stencil) in such extension. As I know, these two methods are equivalent as long as the derivatives in the PDEs are well approximated.
For your second eqestion. In book `Numerical grid generation, foundations and applications' by Thompson et al., there is an argument about the the influence of the truncation errors of the grid generation on the numerical errors, where his claim that there was no influence was ture because some fortunate cancellations happened in the transform. Generally speaking, it is the solution of the PDE on the grid that is important. Therefore truncation error has to be evaluated for the PDE, not for the separate derivatives. The highorder grid transformation also has to be examined with this criterion, though they are qualitied in most cases. 
Re: What's the future use for highorder END schem
From unsteady inviscid flow with shock present is clear that the latest versions of ENO (WENO,MPWENO) are highly recomendable. However as you increase the order of the polynomials, you need to increase the order of the usual RK scheme to advance the flow to keep the scheme stable and in some sense coherent, this largely increases your CPU time for complex problems (too much for 3D???). Other approaches as ADERm (Toro and Titarev) seems more promising as they avoid RK.
If you are dealing with viscous flows with shocks interactions involved the thing gets more complex and the highest order usually drops to 4 (even when doing DNS). 
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