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AREPO – V. Springel

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arXiv:0901.4107

Adaptive, moving, unstructured hydrodynamics, locally adaptive time-steps, self-gravity + Galilean Invariance

i.e. Everything you ever wanted except MHD ;)

66 journal pages!

- Why do we want/need all these features?
- Unstructured grid: adapt to needs of the problem
- Efficiency concern

- Adaptive grid: put in more resolution where necessary
- Accuracy concern

- Moving grid: follow the flow and place computation where it needs to be
- Accuracy and efficiency concerns

- Unstructured grid: adapt to needs of the problem

- Moving grids are nothing new, developed extensively in 1970s
- Fundamental limit has always been mesh entanglement
- Mesh can become “over”-distorted or cells virtually degenerate
- Either stop, or resort to some other method (mapping back to regular grid)

Circumcircle does not enclose

any other vertices.

Form usual state vector, flux function & Euler (conservation) equations

Fluid state described by cell averages

Use Euler equations + convert volume integral to surface integrals

w cell boundary velocity, w=0 for Eulerian code

Moving grids won’t follow flow perfectly so still need to include w term

Using Aij to describe orientation of faces

MUSCL-Hancock scheme

Unsplit – all fluxes computed in one step

Green-Gauss theorem over faces is inaccurate

Use a more complex construction

Where cij is vector to the centre of mass of face

e.g. construct density at a point by

Maintains second order accuracy in smooth regions

Apply slope limiter as well

It’s 1:07 am...

Simplest approach is to simply follow fluid speed of cell

Can lead to poor cell aspect ratios

- Iterate the mesh generation points to better positions
- Lloyd’s Algorithm:
- Move mesh generation points to the centre of mass of their cell
- Reconstruct Voronoi tessellation
- Repeat

- Net effect is mesh relaxes to a “rounder” more regular state

Original distribution of cells

After 50 iterations of Lloyd’s algorithm

- Add velocity adjustment to move mesh generation point towards centre of mass
- Basically:
- Calculate volume of cell & centre of mass
- Associate effective radius with this volume R
- If centre of mass exceeds some set fraction of R, add component to move mesh generation point toward COM
- True method softens point from where there is no correction to a full correction enforced

- No hierarchy of grids
- Just add points or remove as necesary
- However, not really a significant part of the algorithm
- Moving grid covers main adaptive aspects

Treats cells as top-hat spheres of constant density

Force softening is applied but not actually necessary on the grids (cells maintain very regular spacing)

Carefully applied a correction force arising from different force softenings associated with each cell

1-d acoustic wave evolution

Sod shock

Interacting blast waves

Point explosion (i.e. Sedov-like test)

Gresho vortex problem

Noh shock test

KH instability

RT instability

Stirring test

Fixed Moving

Moving grid seems to handle contact discontinuity slightly better

No surprises here

IGNORE the red line on the plots ppt screwed up

At simulation time t=2.0

At simulation time t=2.0 – more mixing in the fixed mesh!

Solution becomes dominated by advection errors

Moving mesh solution is said to be “identical” regardless of v

Moving mesh

Fixed mesh

Moving mesh

Fixed mesh

Evrard collapse test (spherical collapse of self-gravitating sphere)

Zel’dovich pancake (1-d collapse of a single wave but followed in 2-d)

The “Santa Barbara” cluster (cosmological volume simulated with adiabatic physics)

Galaxy collision

“Trivial” problem of collapsing sphere of gas

Accretion shock is generated

Common test for self-grav hydro codes

- Cosmological simulation of one large galaxy cluster, large comparison project in 1999
- Showed a number of differences between codes

- Self gravitating adiabatic perfect gas + dark matter problem
- Consistently shown differences in behaviour in cores of clusters
- Very important to estimates of X-ray luminosity

Dark matter calculations very

close – thank goodness

Some significant differences

(residual would have been nice)

Appear closer than temps

Entropy profile hints at a core

For 1283 run

I can’t find any!

One suspects that the method might be somewhat slow at the moment

Probably not a bad thing right now – most of the computations are linear algebra on small matrices

Can decompose the problem well enough to keep parallel computers very busy...

- Simply amazing collection of features
- the $64,000 is not answered – how fast does it run?
- Memory efficiency is not great...

- BUT! Mesh entanglement problem solved
- Derefining problem solved
- Errors on most problems exceptionally well behaved