When I compute a blunt body at high speed with Roe's flux difference and MUSCL method in implicit scheme,I find the famous carbuncle phenomenon regarding with any my known entropy fixes including Harten-Yee's,chakravarthy's and Harten-Hymen's entropy fixes etc.I am exhausted to do with it.

Additionally,when I use the subiteration method or dual time stepping to compute some simple unsteady shock moving problem,I find the loss of time accuracy as compared with the theoretical or experiment result unless increase the number of subiterations.I don't know whether has a criterion about the number of subiterations in order to preserve the time accuracy.

Could someone give me some useful suggestions about them? Thank you!

I have heard about those names you mentioned, but I am not familiar with the detail scheme or methods. I think, if you are having problems with numerical soutions of fluid dynamic flows, then there are only two possible reasons for that. One is that the mesh is not fine enough. The other is that the method used can not simulate the physics of the flow. For example, the shock wave is basically a discontinuity (it is possible to solve for the one-D shock structure, but normally, it is very thin under the conditions we encounter .),so, the so-called shock captured methods or results basically are not physically or mathematicall correct. If it is a shock, then a shock fitting method is more appropriate, and you can do it iteratively. Trying to fix the shock capturing methods or solutions is only going to cost you a lot of time. Regardless of what you do with the shock capturing methods, the results are not going to be right.( it can only look better than other similar methods) My suggestion is that try to understand the physics of the fluid dynamics and the traditional numerical methods, and see what you can do with the CFD. A seemingly correct solution with a lot of fixes eventually will create more problems later on. ( I think, this world will become more peaceful when people are not allowed to use their names. This also applies to the current wars in Europe. Since it is impossible to remove a name from one's brain, the result is normally either to eliminate the name from existence or to be eliminated. The end result is the same: to erase a name from the brain. ) So, next time, please tell me what the method does, instead of the name associated with it.

A few comments:

(1) The carbuncle phenomenon can be really annoying. In order to get rid of it using a Roe-type scheme, you will probably have to play with increasing the entropy damping coefficients, which can be very problematic. You may want to try using another flux scheme ( such as Van Leer's Flux Vector Splitting or Liou's AUSM method), they are not generally as prone to the carbuncle problem.

For some cases, especially if you are using a stretched viscous grid, local time stepping can cause oscillations in the vicinity of the stag pt that do not readily damp out. For these cases, I have found I can get good, converged answers by running with a uniform global time step. It may take longer, but sometimes its the only way to get a solution. Lastly, here are 2 references that may help- Gaitonde,D., and Shang, J.S., "Accuracy of Flux Split Algorithms in High-Speed Viscous Flows", AIAA Journal, July 1993 (Vol. 31, No. 7), p.1215. AIAA 92-0545 by Grant Palmer, "Effective Treatment of the Singular Line Problem for Three Dimensional Grids"

2. The accuracy of your unsteady shock calculations is directly related to several things a) the order of accuracy of your time differencing (ie. Euler implicit is first-order time accurate, trap rule is second order, etc.), b) any factorization error associated with your time integration scheme, and c) the size of your time step.

Using subiterations basically helps remove some of the factorization error associated with single step methods ( such as the error associated with a Beam-Warming type approximate factorization). Usually, solving the inner iteration problem until the linear problem residual is decreased by two to three orders of magnitude should improve your time accuracy. Of course, all this is predicated on the assumption that your time step is not too large to give accurate results. If you are using a first-order method with large time steps, it can really smear things out.

Hope this helps!

A brief remark regading the entropy fix:

It also depends on which characteristic fields one applies the entropy fix. Theoretically the fix is only required in the non-linear fields with eigenvalues 'lambda+u' and 'lambda-u'. But for this kind of problem in hypersonics it is also important (I think so) to fix the linear degenerated fields with eigenvalues 'u'. This puts extra dissipation to the whole scheme and apparently damps out those oszillations. Up to my knowledge the problem is that Roe' solver cannot treat shears very well (at least not 100% correct), and the problem might be associated with this insuffiency. If one modifies the linear degenerated fields, one has to adjust the delta-parameter according to the local flow speed or mach-number: So, at the shock one gets a high value, in the boundary layer (for Navier-Stokes) it becomes low, as it should be. Otherwise one destroys the boundary layer, since adding a fix to the degenerated fields is not good for getting shears right.

I hope that helps a bit. I actually observe those kinds of wiggles in sub-sonic flows at the nose of an airfoil as well, but I am not sure if this is the same 'feature' of the riemann solver or not ?! If someone has observed those wiggles for profils in sub or transsonic flows I would be interested in.

Frank Bramkamp