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AREPO – V. Springel. 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!. AREPO – V. Springel. Why do we want/need all these features?

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AREPO – V. Springel

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Arepo v springel

AREPO – V. Springel

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!


Arepo v springel1

AREPO – V. Springel

  • 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


History moving meshes

History: Moving Meshes

  • 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)


Delaunay voronoi tessellations

Delaunay & Voronoi tessellations

Circumcircle does not enclose

any other vertices.


Hydro formulation

Hydro formulation

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


Finite volume method

Finite-volume method

Fluid state described by cell averages

Use Euler equations + convert volume integral to surface integrals

w cell boundary velocity, w=0 for Eulerian code


Can t guarantee w v

Can’t guarantee w=v

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

Using Aij to describe orientation of faces


Riemann problem step

Riemann problem step

MUSCL-Hancock scheme

Unsplit – all fluxes computed in one step


Gradient construction

Gradient construction

Green-Gauss theorem over faces is inaccurate

Use a more complex construction

Where cij is vector to the centre of mass of face


Linear reconstruction

Linear reconstruction

e.g. construct density at a point by

Maintains second order accuracy in smooth regions

Apply slope limiter as well


Riemann solver

Riemann solver

It’s 1:07 am...


Mesh movement criterion

Mesh movement criterion

Simplest approach is to simply follow fluid speed of cell

Can lead to poor cell aspect ratios


Solving the mesh movement problem

Solving the mesh movement problem

  • 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


Example

Example

Original distribution of cells

After 50 iterations of Lloyd’s algorithm


Mesh movement criterion ii

Mesh movement criterion II

  • 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


Comparison on sedov test

Comparison on Sedov test


Refining derefining

Refining & derefining

  • 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


Timestepping

Timestepping


Gravity calculation

Gravity calculation

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


Pure hydro test cases

Pure hydro test cases

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


Sod shock

Sod shock

Fixed Moving

Moving grid seems to handle contact discontinuity slightly better

No surprises here

IGNORE the red line on the plots ppt screwed up


Kh instability results fixed mesh

KH instability results: fixed mesh

At simulation time t=2.0


Kh instability results moving mesh

KH instability results: moving mesh


Kh movie

KH movie


Khi at t 2 0

KHI at t=2.0

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


Khi with boosts fixed mesh

KHI with boosts (fixed mesh)

Solution becomes dominated by advection errors

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


Rayleigh taylor instability

Rayleigh Taylor Instability

Moving mesh

Fixed mesh


Rt with boosts

RT with boosts

Moving mesh

Fixed mesh


Examples with self gravity

Examples with self-gravity

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


Evrard collapse

Evrard Collapse

“Trivial” problem of collapsing sphere of gas

Accretion shock is generated

Common test for self-grav hydro codes


Energy profile

Energy profile


Santa barbara cluster

“Santa Barbara” cluster

  • 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


Radial profiles

Radial profiles

Dark matter calculations very

close – thank goodness

Some significant differences

(residual would have been nice)


Radial profiles1

Radial profiles

Appear closer than temps

Entropy profile hints at a core

For 1283 run


Rotation test movie

Rotation test movie


Timing figures

Timing figures?

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...


Summary

Summary

  • 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


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