I am trying to compte a 2-D MHD steady state solution using an upwind scheme and a multigrid space domain.

I don't really understand how to compte the steady state with the multi-stage runge-kutta scheme using a local time step for each cell.

When we compute the solution for a cell at a local time t we use its neighbors to evaluate the cell residual. But the parameters of these neighbors cells are obtained for another local times different of the local time t (since we are using diffrent time stepping in function of the volume of each cell). I tried to do a linear extrapolation of these parameters from the local times of their cells to the local time t (using the last known residual for each neighbor cell). This method didn't work because the residual is not linear.

Is there any other method to do that ?

Thanks Mehdi

I am not very familiar with MHD. But it seems that the problem is more related to numerics.

A Runge-Kutta scheme is an explicit scheme. For stability reasons, the time-step has to be limited by a local or global criteria.

Computing a steady-state solution is equivalent to compute the solution at large physical times. Using a local time-step for each cell is equivalent to take the larger numerically-stable time-step for each cell, thus "bringing cell as close as possible to the steady-state independantly of the others".

In theory, you don't have to extrapolate in time the variables in the neighbouring cells, eg construct the RHS as if all cells were at the same physical time. This should work since you are only interested by the steady-state solution, when the variables are not evolving anymore.

For pure fluid dynamic, implicit methods are prefered since larger "time-steps" can be used and steady-state solution is reached quicker.

Thank you for you answer JDC, I think that you understood perfectly my problem.

I tried first you method : computing the Residual using the parameters of the neighbors cell without taking into account their local time. This method gave me a lot of problem especially during the reconstruction process (negative pressure or negative mass denstities).

I will try it again and see what can happen and let you know if it works.

Thank you anyway for you help JDC