Hi,

I am trying to code an Adaptative upwind Scheme for Ideal MHD. I have a question concerning the Local time steppping for the multiscale mesh.

A 2 Stage Runge Kutta classic scheme for a time accurate can be written as U0=Un U1=U0+a1*Dt*Res(U0) U2=U0+a2*Dt*Res(U1) Un+1=U2

where a1 and a2 are the multistage cofficients and Dt the global time step.

If we start to use a locat time stepping, every cell should converge with its own speed. So every cell in the grid has its own local time. How we can apply the Runge Kutta scheme for a given cell using to compute its neigbors who are in other local times ?

The classic range Kutta sheme suppose that to compute the Un+1 we will use the cell at the time n and all the neighbors cell (to compute the residu, at the same time too).

Thank you

Local time stepping is meant for steady state calculation. When computing Res(Q*_i-1,Q*_i,Q*_i+1), say, Q*_i-1 and Q*_i+1 in other cells are just a relaxed values, which are relaxed from another local time in previous iteration "*". But this does not enfluence the final solution of steady state.