Rubens Machado

Seminário de Extragaláctica - IAG 27/10/2011

AREPO: a moving-mesh code for cosmological hydrodynamical simulations

E pur si muove: Galiliean-invariantcosmological hydrodynamical simulations

on a moving mesh

Springel, 2010 arxiv:0901.4107

● Moving mesh cosmology: numerical techniques and global statistics (Vogelsberger et al, 2011) arxiv:1109.1281

● Moving mesh cosmology: the hydrodynamics of galaxy formation (Sijacki et al, 2011) arxiv:1109.3468

● Moving mesh cosmology: characteristics of galaxies and haloes (Keres et al, 2011) arxiv:1109.4638

and some of its recent results

collisionless simulations

(only dark matter)

● N-body problem is well understood

● there is little doubt about what is required to achieve high accuracy

hydrodynamical simulations

● variety of fundamentally different numerical methods in use

● most prominent: - Lagrangian (particles) - Eulerian (mesh)

● both have problems that make them innacurate in certain regimes

slide from Springel, 2006

slide from Springel, 2006

mesh SPH

● SPH smooths out sharp shocks and contact discontinuities

● self-gravity needs to be done on a mesh

discrete distribution of particles

discrete distribution of particles

What is the density at any given location?

+

Smoothed Particle Hydrodynamics (SPH)

kernel interpolationto build continuous fluid quantities from discrete tracer particles+

Smoothed Particle Hydrodynamics (SPH)

kernel interpolationto build continuous fluid quantities from discrete tracer particles+

Adaptive Mesh Refinement (AMR)

Adaptive Mesh Refinement (AMR)

RAMSES, Teyssier 2010

large dynamical range of cosmological simulations:dense regions require high resolution; low resolution for empty regions

AREPO: based on a moving unstructured mesh

● mesh defined by Voronoi tesselation of a set of discrete points

● grid-generating points move with the flow

● inherits main advantages - from SPH: resolution follows density

automatically and continuously - from Eulerian codes: finite volume discretization gives accurate treatment of instabilities

● avoids main problems

- from SPH: noise and diffusiveness - from Eulerian: lack of Galilean-invariance

Tesselationthe careful juxtaposition of shapes in a mosaic pattern

no overlaps, no gaps

Tesselationthe careful juxtaposition of shapes in a mosaic pattern

no overlaps, no gaps

Delaunay Tesselationtriangulation of the plane

● grid-generating points: vertices of triangles● maximizes smallest angle● within each circumference: no other points

Delaunay Tesselation

centers of circumferences

connecting the centers:

Voronoi Tesselation

The centers of the circumcircles around each Delaunay triangle define the vertices of the Voronoi cells.

Voronoi Tesselation

mesh-generating points

Voronoi Delaunay both

- calculate a new Voronoi tesselation, based on current coordinates of mesh-generating points

- calculate (density, velocity, pressure) in each cell

- assign velocities to the mesh-generating points

- compute flux across each Voronoi cell

- update quantities for this timestep

- move mesh-generating points

at each timestep:

Mesh regularization

for computational efficiency,regions of similar gas properties should be representedby cells of comparable size

desirable to have cells wherecenter-of-mass is close to itsmesh-generating point

Mesh regularization

Adaptive mesh refinement no longer necessary:

The resolution automatically stays where it is needed

Self-gravity ● in cosmological simulations,dark matter structures growfrom very small seed perturbations

● in AMR codes, grid refinementmay be placed too late

Some test problems: Kevin-Helmholtz instability

Some test problems: Kevin-Helmholtz instability

AREPO moving AREPO fixed ATHENA

(in the non-linear regime, KH instability seems to develop faster with AREPO moving mesh)

Some test problems: Kevin-Helmholtz instability

AREPO fixed = ATHENAAREPO fixed = ATHENAAREPO fixed = ATHENA

v = 1 V = 10 v = 100Frame of reference velocity:

with AREPO moving, result is independent of frame-of-reference

Some test problems: isolated disk galaxy

Some test problems: galaxy collision

initial conditions

Some test problems: galaxy collision

AMR not well suited for this problem (high resolution regions moving; no Galilean invariance)

Some test problems: galaxy collision

starsgas

___ AREPO- - - GADGET2

Speed:● 2x slower than SPH at same resolution● 3-4x slower than Eulerian fixed mesh (polyhedra with more than 6 faces)

● once self-gravity is included: performance difference less important

● AREPO reaches good accuracy in test problems at lower resolution than SPH and fixed mesh

Paralelization of the tesselation:

● Point set is decomposed into disjoint spacial domains, each mapped to a different compute core with its own physical memory ...

- first hydrodynamical simulations of galaxy formation using AREPO

- comparison with GADGET (same gravity solver, same star formation treatment)

global baryon statistics with AREPO:

- lower mean temperatures- reduced amount of hot gas- more gas cooling at low z- higher star formation rates in late times

- overall distribution of gas temperature and density broadly in agreement

- SFR at high z in good agreement

Gas density

Code performance

AREPO

GADGET

- With AREPO, gas from infalling substructures is readily depleted and incorporated into the host halo atmosphere, facilitating the formation of an extended central disk.

- With GADGET, gaseous subclumps are more coherent, transforming the central disk as they impact it.

- variety of numerical experiments to establish link between simple problems with analytic solutions and systematic effects in cosmological simulations of galaxy formation

gas density1014 Mo isolated halo + 10 orbiting subhalos

AREPO:

http://www.mpa-garching.mpg.de/~volker/arepo/

Moving Mesh Cosmology:

http://www.cfa.harvard.edu/itc/research/movingmeshcosmology/

"Sator Arepo Tenet Opera Rotas"