modélisation numérique multi-échelle des écoulements mhd en astrophysique

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)


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


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


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


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


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


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


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

?

?? ?


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.


Field loop advection test (Gardiner & Stone 2005)


Current sheet and magnetic reconnection


ABC flow and the fast dynamo: towards Rm=106 ?

Galloway&Frisch (1986)

Lau&Finn (1993)

323 643

1283 2563


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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


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.


Conclusion and perspectives

modélisation numérique multi-échelle des écoulements mhd en astrophysique

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