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1Séminaire AIM (20/06/06) Romain Teyssier
Modélisation numérique multi-échelledes écoulements MHD en astrophysique
Romain Teyssier (CEA Saclay)
Sébastien Fromang (Oxford)
Emmanuel Dormy (ENS Paris)Patrick Hennebelle (ENS Paris)
François Bouchut (ENS Paris)
2Séminaire 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
3Séminaire 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. 2005
• 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
4Séminaire 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
5Séminaire 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 tn 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 tn+1/2 in each normal direction
Corner Transport Upwind
Predictor step in transverse direction only
Flux functions computed using 1D Riemann problem at time tn+1/2 in each normal direction
6Séminaire AIM (20/06/06) Romain Teyssier
For piecewise constant initial data, theflux 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
7Séminaire 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
8Séminaire 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
?
?? ?
9Séminaire AIM (20/06/06) Romain Teyssier
n=400
Compound wave (Torrilhon 2004)
n=800
n=20000
neff=106
: 2 solutions: 2 shocks or 1 c.w.: 2 shocks onlyDissipation properties are crucial.Only AMR can resolve scales small enough within reasonable CPU time.
10Séminaire AIM (20/06/06) Romain Teyssier
Field loop advection test (Gardiner & Stone 2005)
11Séminaire AIM (20/06/06) Romain Teyssier
Current sheet and magnetic reconnection
12Séminaire AIM (20/06/06) Romain Teyssier
ABC flow and the fast dynamo: towards Rm=106 ?
Galloway&Frisch (1986)
Lau&Finn (1993)
323 643
1283 2563
13Séminaire AIM (20/06/06) Romain Teyssier
QuickTime™ et undécompresseur codec YUV420
sont requis pour visionner cette image.
QuickTime™ et undécompresseur codec YUV420
sont requis pour visionner cette image.
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 transfer2- fragmentation (binary formation)3- jets and outflows
Face-on
Bz=0
Side-on
Face-on
M/=2
Side-on
14Séminaire AIM (20/06/06) Romain Teyssier
Details in the outflow structure
Lax-Friedrich Riemann solver Roe Riemann solver
Conical jet (Roe) versus cylindrical jet (Lax-Friedrich) ?Sensitive to small-scale (numerical) dissipation.
15Séminaire AIM (20/06/06) Romain Teyssier
Conclusion and perspectives