AREPO – V. Springel

1. 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!

2. 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

3. 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)

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

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

6. 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

7. 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

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

9. 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

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

11. Riemann solver It’s 1:07 am...

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

13. 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

14. Example Original distribution of cells After 50 iterations of Lloyd’s algorithm

15. 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

16. Comparison on Sedov test

17. 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

18. Timestepping

19. 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

20. 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

21. 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

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

23. KH instability results: moving mesh

24. KH movie

25. KHI at t=2.0 At simulation time t=2.0 – more mixing in the fixed mesh!

26. KHI with boosts (fixed mesh) Solution becomes dominated by advection errors Moving mesh solution is said to be “identical” regardless of v

27. Rayleigh Taylor Instability Moving mesh Fixed mesh

28. RT with boosts Moving mesh Fixed mesh

29. 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

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

31. Energy profile

32. “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

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

34. Radial profiles Appear closer than temps Entropy profile hints at a core For 1283 run

35. Rotation test movie

36. 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...

37. 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