I am coding an upwind scheme with adaptative mesh to solve an MHD problem with a strong shock. I am using a 2 stage Runge-Kutta scheme for stepping in time.

I have problem to converge, since my solution start to oscillate very quickly (5 or 6 iterations), the osciallation become verry strong and crash the solution.

These oscillation seems to come from the shock and expand to all the space domain. The oscialltion frequency is the same of the grid, so refining more does'nt change resolve the problem.

- I want to know what kind of solution I have to damp such oscialltion and make my solution converge. - Should I add a terme in d2_U/d_x to add an actificial viscosity in my solver ? What consequence it will have on the final solution.

Thank you

I assume you are using higher order upwinding. Upwind schemes are subject to oscillations near shocks. You can use some limiter ( e.g. minmod, vanleer..) to restrit the scheme to first order near shocks to make the scheme more dissipative and avoid oscillations.

Typically, they add artificial viscosity for central difference schemes as it has no dissipation..( so the term artificial or numerical viscosity)

I hope this helps.

Ravi

Thank you for you answer Ravi

In fact, I am using a second order in space and time upwind MUSCL scheme. during the reconstruction I use a minmod type diffusive limiter to reconstruct the cell's edges variables (Barth limiter). Despite I am forcing this limiter to 0 ( forcing the scheme to be just a first order in space all the time), I still have the same oscillations that are coming form the shock.

Turning on or off the grid refinement doesn't change anything too as I said.

How can I remove such oscillation that make my solution diverge ?

PS : I am simulating a supersonci plasma shock (M=10)

Did you experience the so called carbuncle artifact for blunt body in supersonics? there are much work done on it. You can see recent (2002-2003) JCP paper. It seems that all 1D-based Riemann -type scheme will face such "carbuncle artifact" in hypersonics.

Always depend on the Riemann Solver you are using, Roe's solver and HLL will have carbuncle phenomena and do not predict contact waves. Use HLLC or HLLEM, those are not affected by those phenomena and they are not that much expensive. Anyway using MHD, the Riemann Solver must be different so the problem must be somewhere else (I cannot coment on that) I rember peope dong it have to put artifcats for div B=0 buy I really am not an expert on it. Using MUSCL or other limiter, the damping should be effective (turn MUSCL off) and you shouldn't see the oscillations (first order is quite dissipative).

If you are not using Rieamnn solvers, what are you using?, do you have viscoisty ? or walls? does it work if you turn the lectric/magnetic force off (not solving the M in MHD