Modlisation numrique multichelle des coulements MHD en astrophysique

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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)
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Les équations de la MHD idéale

• Conservation de la masse
• Conservation de la quantité de mouvement
• Conservation de lénergie
• Conservation du flux magnétique
• Pression totale
• Energie totale

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Godunov method and MHD

• 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
• Finite volume and cell-centered schemes
• div B cleaning using Poisson solver
• div B waves (Powells 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
  BalsaraSpicer 99
• 2D Riemann solver to compute EMF
  LondrilloDelZanna 01,05 Ziegler 04,05
• High-order extension of Balsaras 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,
  AA, in press
• Applications Kinematic Dynamos and astrophysical
  MHD

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Godunov method for 1D Euler systems
Finite volumes conservation laws in integral form

• Piecewise constant initial states
• self-similar Riemann solution

Modified equation has diffusion term
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2D schemes for Euler systems
2D Euler system in integral form

• 2D Riemann problems
• ? self-similar (exact ?) solution relative to
  corner points
• Flux function is not self-similar (line
  averaging) ? predictor-corrector schemes ?

• 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 tn1/2 in each normal
direction
Corner Transport Upwind Predictor step in
transverse direction only Flux functions computed
using 1D Riemann problem at time tn1/2 in each
normal direction
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The induction equation in 2D
Finite-surface approximation (Constrained
Transport) Integral form using Stokes
theorem

• For piecewise constant initial data, the
• flux function is self-similar at corner points

For pure induction, the 2D Riemann problem has
the following exact (upwind) solution Numeric
al diffusivity and
Induction Riemann problem
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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
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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
?
?
?
?
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Compound wave (Torrilhon 2004)
??? 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.
neff106
n400
n800
n20000
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Field loop advection test (Gardiner Stone 2005)
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Current sheet and magnetic reconnection
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ABC flow and the fast dynamo towards Rm106 ?
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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 theory1- angular
momentum transfer2- fragmentation (binary
formation)3- jets and outflows
Face-onBz0Side-on
Face-onM/?2Side-on
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Details in the outflow structure
Conical jet (Roe) versus cylindrical jet
(Lax-Friedrich) ?Sensitive to small-scale
(numerical) dissipation.
Lax-Friedrich Riemann solver
Roe Riemann solver
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Conclusion and perspectives