I want to simulate numerically the Reactive Euler equations for supersonic flow with Finite Differences. I have thought of MacCormack time-dependent algorithm, but it seems to me a bit old, and very oscillating.

Do you know some modern scheme that belongs to the state of the art of CFD and doesn't have any stability constraint or oscillation problems?.

I only need some names or points to begin to search for it at the library.

Thanks in advance

The first choice for shock problems is TVD or Godunov. But TVD scheme is too diffusive in boundary layer. Godunov is of higher resolution, but more complex & may also produce artifacts in shear layer. The reaction adds chemical scales, which makes necessary to use higher order or clearer schemes. PPM, ENO, WENO, CESE, fluctuation spliting, are desirable. But Some of them are not as robust as the Harten-Yee upwind TVD. I prefer to first use the conventional Harten upwind TVD, then use other particular higher oder schemes with precaution that may fails in some cases.

The MacCormack sheme can be prformed by the artificial viscoities terms (the 2ed and 4th ordre)for eliminate the anscillations near the schoc.

Depends of type of flow Godunov is a good choice if you are dealing with laminar flows, use it together with Approximate Riemann-Solvers such as HLLEM or HLLC. In laminar flow a second order extension such TVD-MUSCL, or whaetver choice should be enough (even in shear/boundary layers) For the chemistry part you probably have to use an implicit solver because chemical time-scales will be smaller (in general) than flow scales. If your flow is turbulent you may want to be more careful with the numerics, TVD schemes revert to first order near discontinuities and may disipate small-scale turbulence.

Thanks to all replies.

If you want to add further information, I'm trying to simulate a RAM accelerator device, both subdetonative and superdetonative modes, with Euler Reactive equations, and with a simple one step chemical reaction of combustion.

I hope it helps you in your advices.

As far as I know, it is impossible to have turbulent flow while employing Euler equations (???)

Then Godunov with HLLC (or HLLEM) and TVD extension would be my choice, you are solving Euler so you do not have boundary layers nor turbulence so your main concern is basicallyshock waves. Finite Volume Godunov's schemes are the best when dealing with shocks. Check the books by Laney (2000?) in CUP and Toro (200? Springer-Verlag).

Probably an operator spliiting for the chemical part will do fine. If you are using one step you may not even needed.

u can use van leer flux vector splitting scheme to discretize inviscid fluxes. it is ver robust and comparatively more stable than Roe scheme for reactive problems. can i know further details on ur problem such as Mach number range, implicit/explicit, geometry as i have coded for equilibrium reacting air with 5 species model.

It is good to choose two schemes below.

(1) AUSM+ or (2) AUSM-DV

These schemes can compute a two or more Mach flow stably.

If you want to compute around Mach=1 only,you can use "Roe scheme + MUSCL interpolation".

I'm sorry to be stragne English.