Dear Sir,

I have a question regarding the behaviour of numerical schemes, in particular Roe's scheme for evaluating Inviscid fluxes.

I have a NS-code which works for supersonic and hypersonic flows with implicit time marching (euler). I have noticed that though for High Mach numbers the residue oscillates and I have to reduce the courant number, though implicit schemes are unconditionaly stable and that it should converge for any courant/CFD number. I would like this point to be clarified.

thanks, ravi.

Unconditional stability does not mean the scheme is always stable under any numerical or flow conditions, since it is based on linear stability analysis.

First-order Euler method is known as fairly stable method, however, it does not allow very large CFL number also. If your problem is really stiff in time, you can use other implicit scheme. But, in my opinion, they need exessive computation time.

There can be several reasons for that your solution does not converge for high CFL number : choice of grid system, approximation of inviscid/viscous flux jacobian, using factorised scheme or not, etc.

Hi, As for the theory it is ok to say that Implicit schems are not having the stability problem and also no CFL restriction. but the question is whether you are appling boundary conditions implicitly or not. boundary conditions in the case of euler equations is very easy. but for Navier strokes it is difficult(MacCormak paper 1982).

as u are saying there are oscilations in residual u have to check whether you are appling sufficient numerical diffusion. when u deal with NS equations u have to add second order diffusion implicitly and 4th order diffusion explicitly. these conditions may not be nessary for every case. these are required for flows which are having shocks etc. so there are lot of things u have to take care. Godunov schems are good choice. by Rajani Kumar

Hi Rajani Kumar,

Thanks for the wealth of info. on schemes. Indeed, I haven't applied the boundary condition IMPLICITLY and it's one of the reasons that it behaves this way. Apart from that I have blowing at the plate surface and residues oscillate more vigourously in this case.

Shocks do are present in this case as I am trying to solve for supersonic and hypersonic flows with blowing/gas injection at the surface. When there is no blowing the residue fall is quite smooth.

By the way let me know your email ID. Are you a student studying in any Univ. in US. Thanks for the info again.

ravi.