1 Séminaire AIM (20/06/06)Romain Teyssier Modélisation numérique multi-échelle des écoulements MHD en astrophysique Romain Teyssier (CEA Saclay) Sébastien Fromang (Oxford) Emmanuel Dormy (ENS Paris) Patrick Hennebelle (ENS Paris) François Bouchut (ENS Paris)

2 Séminaire AIM (20/06/06)Romain Teyssier Les équations de la MHD idéale Conservation de la masse Conservation de la quantité de mouvement Conservation de l’énergie Conservation du flux magnétique Pression totale Energie totale

3 Séminaire AIM (20/06/06)Romain Teyssier Euler equations using finite volumes: decades of experience in robust advection & shock-capturing schemes Godunov; MUSCL (Van Leer); PPM (Woodward & Colella) Toro 1997 Ideal MHD : Euler system augmented by the induction equation 1.Finite volume and cell-centered schemes –div B cleaning using Poisson solver –div B waves (Powell’s 8 waves formulation) –div B damping Crockett et al Constrained Transport & staggered grid (Yee 66; Evans & Hawley 88) –1D Godunov fluxes to compute EMF Balsara&Spicer 99 –2D Riemann solver to compute EMF Londrillo&DelZanna 01,05; Ziegler 04,05 –High-order extension of Balsara’s scheme Gardiner & Stone 05 Our goal: design fast, second-order accurate, Godunov-type, for a tree-based AMR scheme with Constrained Transport Teyssier, Fromang & Dormy 2006, JCP, in press Fromang, Hennebelle & Teyssier 2006, A&A, in press Applications: Kinematic Dynamos and astrophysical MHD Godunov method and MHD

4 Séminaire AIM (20/06/06)Romain Teyssier Godunov method for 1D Euler systems Piecewise constant initial states: self-similar Riemann solution Finite volumes: conservation laws in integral form Modified equation has diffusion term

5 Séminaire AIM (20/06/06)Romain Teyssier 2D Riemann problems:  self-similar (exact ?) solution relative to corner points Flux function is not self-similar (line averaging)  predictor-corrector schemes ? 2D schemes for Euler systems 2D Euler system in integral form: Godunov scheme No predictor step. Flux functions computed using 1D Riemann problem at time t n in each normal direction. Courant condition: Runge-Kutta scheme Predictor step using Godunov scheme and  t/2 Flux functions computed using 1D Riemann problem at time t n+1/2 in each normal direction Corner Transport Upwind Predictor step in transverse direction only Flux functions computed using 1D Riemann problem at time t n+1/2 in each normal direction

6 Séminaire AIM (20/06/06)Romain Teyssier For piecewise constant initial data, the flux function is self-similar at corner points The induction equation in 2D Finite-surface approximation (Constrained Transport) Integral form using Stoke’s theorem For pure induction, the 2D Riemann problem has the following exact (upwind) solution: Numerical diffusivity and Induction Riemann problem

7 Séminaire AIM (20/06/06)Romain Teyssier RAMSES: a tree-based AMR parallel code Fully Threaded Tree (Khokhlov 98) Cartesian mesh refined on a cell by cell basis octs: small grid of 8 cells, pointing towards 1 parent cell 6 neighboring parent cells 8 children octs Coarse-fine boundaries: buffer zone 2-cell thick Time integration using recursive sub-cycling Parallel computing using the MPI library Domain decomposition using « space filling curves » Good scalability up to 4096 processors Euler equations, Poisson equation, PIC module Cooling module, implicit diffusion solver Induction equation Ideal MHD needs 7-wave Riemann solvers: Lax-Friedrich and Roe

8 Séminaire AIM (20/06/06)Romain Teyssier AMR and Constrained Transport « Divergence-free preserving » restriction and prolongation operators Balsara (2001) Toth & Roe (2002) Flux conserving interpolation and averaging within cell faces using TVD slopes in 2 dimensions EMF correction for conservative update at coarse-fine boundaries ? ? ??

9 Séminaire AIM (20/06/06)Romain Teyssier n=400 Compound wave (Torrilhon 2004) n=800 n=20000 n eff =10 6  : 2 solutions: 2 shocks or 1 c.w.  : 2 shocks only Dissipation properties are crucial. Only AMR can resolve scales small enough within reasonable CPU time.

10 Séminaire AIM (20/06/06)Romain Teyssier Field loop advection test (Gardiner & Stone 2005)

11 Séminaire AIM (20/06/06)Romain Teyssier Current sheet and magnetic reconnection

12 Séminaire AIM (20/06/06)Romain Teyssier ABC flow and the fast dynamo: towards R m =10 6 ? Galloway&Frisch (1986) Lau&Finn (1993)

13 Séminaire AIM (20/06/06)Romain Teyssier Magnetized molecular cloud collapse Rotating, magnetized spherical cloud embedded in low density medium. Barotropic equation of state. AMR with 15 to 20 levels of refinements. Questions for star formation theory: 1- angular momentum transfer 2- fragmentation (binary formation) 3- jets and outflows Face-on B z =0 Side-on Face-on M/  =2 Side-on

14 Séminaire AIM (20/06/06)Romain Teyssier Details in the outflow structure Lax-Friedrich Riemann solverRoe Riemann solver Conical jet (Roe) versus cylindrical jet (Lax-Friedrich) ? Sensitive to small-scale (numerical) dissipation.

15 Séminaire AIM (20/06/06)Romain Teyssier Conclusion and perspectives