AREPO – V. Springel Adaptive, moving, unstructured hydrodynamics, locally adaptive time-steps, self-gravity + Galilean Invariance i.e. Everything you ever wanted except MHD ;) 66 journal pages! arXiv:

AREPO – V. Springel Why do we want/need all these features? Why do we want/need all these features? Unstructured grid: adapt to needs of the problem Unstructured grid: adapt to needs of the problem Efficiency concern Efficiency concern Adaptive grid: put in more resolution where necessary Adaptive grid: put in more resolution where necessary Accuracy concern Accuracy concern Moving grid: follow the flow and place computation where it needs to be Moving grid: follow the flow and place computation where it needs to be Accuracy and efficiency concerns Accuracy and efficiency concerns

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

Delaunay & Voronoi tessellations Circumcircle does not enclose any other vertices.

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

Finite-volume method Fluid state described by cell averages Fluid state described by cell averages Use Euler equations + convert volume integral to surface integrals Use Euler equations + convert volume integral to surface integrals w cell boundary velocity, w =0 for Eulerian code w cell boundary velocity, w =0 for Eulerian code

Can’t guarantee w=v Moving grids won’t follow flow perfectly so still need to include w term Moving grids won’t follow flow perfectly so still need to include w term Using A ij to describe orientation of faces Using A ij to describe orientation of faces

Riemann problem step MUSCL- Hancock scheme MUSCL- Hancock scheme Unsplit – all fluxes computed in one step Unsplit – all fluxes computed in one step

Gradient construction Green-Gauss theorem over faces is inaccurate Green-Gauss theorem over faces is inaccurate Use a more complex construction Use a more complex construction Where c ij is vector to the centre of mass of face Where c ij is vector to the centre of mass of face

Linear reconstruction e.g. construct density at a point by e.g. construct density at a point by Maintains second order accuracy in smooth regions Maintains second order accuracy in smooth regions Apply slope limiter as well Apply slope limiter as well

Riemann solver It’s 1:07 am... It’s 1:07 am...

Mesh movement criterion Simplest approach is to simply follow fluid speed of cell Simplest approach is to simply follow fluid speed of cell Can lead to poor cell aspect ratios Can lead to poor cell aspect ratios

Solving the mesh movement problem Iterate the mesh generation points to better positions Iterate the mesh generation points to better positions Lloyd’s Algorithm: Lloyd’s Algorithm: Move mesh generation points to the centre of mass of their cell Move mesh generation points to the centre of mass of their cell Reconstruct Voronoi tessellation Reconstruct Voronoi tessellation Repeat Repeat Net effect is mesh relaxes to a “rounder” more regular state Net effect is mesh relaxes to a “rounder” more regular state

Example Original distribution of cellsAfter 50 iterations of Lloyd’s algorithm

Mesh movement criterion II Add velocity adjustment to move mesh generation point towards centre of mass Add velocity adjustment to move mesh generation point towards centre of mass Basically: Basically: Calculate volume of cell & centre of mass Calculate volume of cell & centre of mass Associate effective radius with this volume R 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 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 True method softens point from where there is no correction to a full correction enforced

Comparison on Sedov test

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

Timestepping

Gravity calculation Treats cells as top-hat spheres of constant density 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) 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 Carefully applied a correction force arising from different force softenings associated with each cell

Pure hydro test cases 1-d acoustic wave evolution 1-d acoustic wave evolution Sod shock Sod shock Interacting blast waves Interacting blast waves Point explosion (i.e. Sedov-like test) Point explosion (i.e. Sedov-like test) Gresho vortex problem Gresho vortex problem Noh shock test Noh shock test KH instability KH instability RT instability RT instability Stirring test Stirring test

Sod shock Moving grid seems to handle contact discontinuity slightly better Moving grid seems to handle contact discontinuity slightly better No surprises here No surprises here IGNORE the red line on the plots ppt screwed up IGNORE the red line on the plots ppt screwed up Fixed Moving

KH instability results: fixed mesh At simulation time t=2.0

KH instability results: moving mesh

KH movie

KHI at t=2.0 At simulation time t=2.0 – more mixing in the 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 Moving mesh Fixed mesh

RT with boosts Moving mesh Fixed mesh

Examples with self-gravity Evrard collapse test (spherical collapse of self- gravitating sphere) Evrard collapse test (spherical collapse of self- gravitating sphere) Zel’dovich pancake (1-d collapse of a single wave but followed in 2-d) 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) The “Santa Barbara” cluster (cosmological volume simulated with adiabatic physics) Galaxy collision Galaxy collision

Evrard Collapse “Trivial” problem of collapsing sphere of gas “Trivial” problem of collapsing sphere of gas Accretion shock is generated Accretion shock is generated Common test for self-grav hydro codes Common test for self-grav hydro codes

Energy profile

“Santa Barbara” cluster Cosmological simulation of one large galaxy cluster, large comparison project in 1999 Cosmological simulation of one large galaxy cluster, large comparison project in 1999 Showed a number of differences between codes Showed a number of differences between codes Self gravitating adiabatic perfect gas + dark matter problem Self gravitating adiabatic perfect gas + dark matter problem Consistently shown differences in behaviour in cores of clusters Consistently shown differences in behaviour in cores of clusters Very important to estimates of X-ray luminosity Very important to estimates of X-ray luminosity

Radial profiles Dark matter calculations very close – thank goodness Some significant differences (residual would have been nice)

Radial profiles Appear closer than tempsEntropy profile hints at a core For run

Rotation test movie

Timing figures? I can’t find any! I can’t find any! One suspects that the method might be somewhat slow at the moment 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 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... Can decompose the problem well enough to keep parallel computers very busy...

Summary Simply amazing collection of features Simply amazing collection of features the $64,000 is not answered – how fast does it run? the $64,000 is not answered – how fast does it run? Memory efficiency is not great... Memory efficiency is not great... BUT! Mesh entanglement problem solved BUT! Mesh entanglement problem solved Derefining problem solved Derefining problem solved Errors on most problems exceptionally well behaved Errors on most problems exceptionally well behaved