AREPO – V. SpringelAREPO – V. Springel

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

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

66 journal pages!66 journal pages!

arXiv:0901.4107

AREPO – V. SpringelAREPO – 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 Unstructured grid: adapt to needs of the problemproblem

Efficiency concernEfficiency concern Adaptive grid: put in more resolution where Adaptive grid: put in more resolution where

necessarynecessary Accuracy concernAccuracy concern

Moving grid: follow the flow and place Moving grid: follow the flow and place computation where it needs to becomputation where it needs to be

Accuracy and efficiency concerns Accuracy and efficiency concerns

History: Moving History: Moving MeshesMeshes

Moving grids are nothing new, Moving grids are nothing new, developed extensively in 1970sdeveloped extensively in 1970s

Fundamental limit has always been Fundamental limit has always been mesh entanglementmesh entanglement Mesh can become “over”-distorted or Mesh can become “over”-distorted or

cells virtually degeneratecells virtually degenerate Either stop, or resort to some other Either stop, or resort to some other

method (mapping back to regular grid)method (mapping back to regular grid)

Delaunay & Voronoi Delaunay & Voronoi tessellationstessellations

Circumcircle does not encloseany other vertices.

Hydro formulationHydro formulation Form usual state vector, flux function Form usual state vector, flux function

& Euler (conservation) equations& Euler (conservation) equations

Finite-volume methodFinite-volume method Fluid state described by cell averagesFluid state described by cell averages

Use Euler equations + convert volume Use Euler equations + convert volume integral to surface integralsintegral to surface integrals

w w cell boundary velocity, cell boundary velocity, ww=0 for =0 for Eulerian codeEulerian code

Can’t guarantee Can’t guarantee w=vw=v Moving grids won’t follow flow Moving grids won’t follow flow

perfectly so still need to include perfectly so still need to include ww termterm

Using Using AAijij to describe orientation of to describe orientation of facesfaces

Riemann problem stepRiemann problem step MUSCL-MUSCL-

Hancock Hancock schemescheme

Unsplit – all Unsplit – all fluxes fluxes computed in computed in one stepone step

Gradient constructionGradient construction Green-Gauss theorem over faces is inaccurateGreen-Gauss theorem over faces is inaccurate

Use a more complex constructionUse a more complex construction

Where cWhere cijij is vector to the centre of mass of is vector to the centre of mass of face face

Linear reconstructionLinear reconstruction e.g. construct density at a point bye.g. construct density at a point by

Maintains second order accuracy in Maintains second order accuracy in smooth regionssmooth regions

Apply slope limiter as well Apply slope limiter as well

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

Mesh movement Mesh movement criterioncriterion

Simplest approach is to simply follow fluid speed Simplest approach is to simply follow fluid speed of cellof cell

Can lead to poor cell aspect ratiosCan lead to poor cell aspect ratios

Solving the mesh Solving the mesh movement problemmovement problem

Iterate the mesh generation points to Iterate the mesh generation points to better positionsbetter positions

Lloyd’s Algorithm:Lloyd’s Algorithm: Move mesh generation points to the Move mesh generation points to the

centre of mass of their cellcentre of mass of their cell Reconstruct Voronoi tessellationReconstruct Voronoi tessellation RepeatRepeat

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

ExampleExample

Original distribution of cellsAfter 50 iterations of Lloyd’s algorithm

Mesh movement Mesh movement criterion IIcriterion II

Add velocity adjustment to move mesh Add velocity adjustment to move mesh generation point towards centre of massgeneration 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 RAssociate effective radius with this volume R If centre of mass exceeds some set fraction If centre of mass exceeds some set fraction

of R, add component to move mesh of R, add component to move mesh generation point toward COMgeneration point toward COM

True method softens point from where there True method softens point from where there is no correction to a full correction enforcedis no correction to a full correction enforced

Comparison on Sedov Comparison on Sedov testtest

Refining & derefiningRefining & derefining No hierarchy of No hierarchy of

gridsgrids Just add points or Just add points or

remove as remove as necesarynecesary

However, not really However, not really a significant part of a significant part of the algorithmthe algorithm

Moving grid covers Moving grid covers main adaptive main adaptive aspectsaspects

TimesteppingTimestepping

Gravity calculationGravity calculation Treats cells as top-hat spheres of Treats cells as top-hat spheres of

constant densityconstant density Force softening is applied but not Force softening is applied but not

actually necessary on the grids (cells actually necessary on the grids (cells maintain very regular spacing)maintain very regular spacing)

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

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

Sod shockSod shock Moving grid Moving grid

seems to seems to handle contact handle contact discontinuity discontinuity slightly betterslightly better

No surprises No surprises herehere

IGNORE the IGNORE the red line on the red line on the plots ppt plots ppt screwed upscrewed up

Fixed Moving

KH instability results: KH instability results: fixed meshfixed mesh

At simulation time t=2.0

KH instability results: KH instability results: moving meshmoving mesh

KH movieKH movie

KHI at t=2.0KHI at t=2.0

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

KHI with boosts (fixed KHI with boosts (fixed mesh)mesh)

Solution becomes dominated by advection errorsMoving mesh solution is said to be “identical” regardless of v

Rayleigh Taylor Rayleigh Taylor InstabilityInstability

Moving mesh

Fixed mesh

RT with boostsRT with boostsMoving mesh

Fixed mesh

Examples with self-Examples with self-gravitygravity

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

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

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

Galaxy collisionGalaxy collision

Evrard CollapseEvrard Collapse ““Trivial” Trivial”

problem of problem of collapsing collapsing sphere of gassphere of gas

Accretion shock Accretion shock is generatedis generated

Common test Common test for self-grav for self-grav hydro codeshydro codes

Energy profileEnergy profile

““Santa Barbara” Santa Barbara” clustercluster

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

Self gravitating adiabatic perfect gas + dark Self gravitating adiabatic perfect gas + dark matter problemmatter problem

Consistently shown differences in behaviour Consistently shown differences in behaviour in cores of clustersin cores of clusters Very important to estimates of X-ray luminosityVery important to estimates of X-ray luminosity

Radial profilesRadial profiles

Dark matter calculations veryclose – thank goodness

Some significant differences(residual would have been nice)

Radial profilesRadial profiles

Appear closer than temps Entropy profile hints at a coreFor 1283 run

Rotation test movieRotation test movie

Timing figures?Timing figures? I can’t find any!I can’t find any! One suspects that the method might One suspects that the method might

be somewhat slow at the momentbe somewhat slow at the moment Probably not a bad thing right now – Probably not a bad thing right now –

most of the computations are linear most of the computations are linear algebra on small matricesalgebra on small matrices

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

SummarySummary Simply amazing collection of featuresSimply amazing collection of features

the $64,000 is not answered – how fast the $64,000 is not answered – how fast does it run?does it run?

Memory efficiency is not great...Memory efficiency is not great... BUT! Mesh entanglement problem BUT! Mesh entanglement problem

solvedsolved Derefining problem solvedDerefining problem solved Errors on most problems exceptionally Errors on most problems exceptionally

well behavedwell behaved