modélisation numérique multi-échelle des écoulements mhd en...

Click here to load reader

Download Modélisation numérique multi-échelle des écoulements MHD en astrophysique

Post on 14-Jan-2016




1 download

Embed Size (px)


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). Les équations de la MHD idéale. Conservation de la masse - PowerPoint PPT Presentation


Présentation PowerPoint - Cosmologie Les grandes structures de l’UniversRomain Teyssier (CEA Saclay)
Conservation de la masse
Conservation de l’énergie
Conservation du flux magnétique
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 (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
Piecewise constant initial states:
integral form
2D Riemann problems:
Flux function is not self-similar (line averaging) predictor-corrector schemes ?
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
For piecewise constant initial data, the
flux function is self-similar at corner points
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
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
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
Lax-Friedrich and Roe
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
: 2 shocks only
Dissipation properties are crucial.
Galloway&Frisch (1986)
Lau&Finn (1993)
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
Lax-Friedrich Riemann solver
Roe Riemann solver
Sensitive to small-scale (numerical) dissipation.