Numerical Methods. (Additional Notes from Talks by PFH)
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1 Numerical Methods (Additional Notes from Talks by PFH)
2 SPH
3 Challenge: POPULAR METHODS FOR HYDRODYNAMICS HAVE PROBLEMS Lucy 77, Gingold & Monaghan 77 Reviews by: Springel 11, Price 12 Smoothed-Particle Hydrodynamics Lagrangian, adaptive, simple, conservative Particles move (freely)
4 Challenge: POPULAR METHODS FOR HYDRODYNAMICS HAVE PROBLEMS Lucy 77, Gingold & Monaghan 77 Reviews by: Springel 11, Price 12 Smoothed-Particle Hydrodynamics No volume partition: point-like particles smoothed into fields [ok in continuum limit ] (average) x x Solve EOM at particle locations (stabilize with artificial diffusion)
5 Challenge: POPULAR METHODS FOR HYDRODYNAMICS HAVE PROBLEMS Keplerian disk/ring (should conserve ICs) Smoothed-Particle Hydrodynamics Lagrangian, adaptive, simple, conservative Artificial diffusion terms: - excess diffusion, viscosity old SPH (Springel 02) (after 20 orbits) new SPH (Hopkins 13) Morris 97, Okamoto 03, Cullen & Dehnen 10, Bauer & Springel 12
6 Challenge: POPULAR METHODS FOR HYDRODYNAMICS HAVE PROBLEMS Traditional SPH GADGET/(old)GASOLINE ~32 neighbors (cubic spline) constant artificial viscosity density formulation Ritchie & Thomas 01, Agertz 07, Price 12, Read 12 Kelvin-Helmholtz Instabilities Modern SPH P-SPH/SPHS/PHANTOM ~ neighbors (alt. kernels) (many people: Read, Dehnen) high-order switches (Cullen+Dehnen) pressure formulation (Hopkins, Saitoh+Makino) artificial diffusion for entropy (Price, Wadsley) old SPH (Springel 02) Sub-sonic turbulence (vorticity) (not SPH!) new SPH (PSPH) (Hopkins 13): >>100 neighbors
7 Grid+boost Rayleigh-Taylor Instabilities MFV MFM SPH: big kernel MFV
8 The Blob (Agertz et al.) MFV MFM
9 everything else MFM P-SPH PSPH TSPH MFM/MFV P-SPH AREPO Sub-Sonic Turbulence (hard for SPH)
10 Artificial Surface Tension (entirely fix-able!) Hopkins 2013
11 Challenge: POPULAR METHODS FOR HYDRODYNAMICS HAVE PROBLEMS Gresho vortex (Dehnen & Aly) Anisotropic Conduction (MTI, HBI, Hall MRI) Fundamental low-order errors: - converge slowly: beat down by increasing kernel size, but this is not efficient! MHD & anisotropic diffusion operators ill-posed 55 neighbors best SPH 100 neighbors normal SPH Dehnen & Aly, Rosswog, Hopkins, Tricco & Price, Read 400 neighbors (not SPH!)
12 AMR + Fixed-Grids
13 Challenge: POPULAR METHODS FOR HYDRODYNAMICS HAVE PROBLEMS Berger & Colella 89 (& others) Reviews by: Teyssier 14 Fixed Adaptive Mesh Refinement Eulerian, well-studied, high-order Each cell carries conserved quantities inside volume Vi Solve Reimann problem between geometric faces
14 Adaptive Mesh Refinement (AMR) CHALLENGE: POPULAR METHODS HAVE PROBLEMS Rayleigh-Taylor instability (AMR, ) Eulerian, well-studied, high-order Excessive mixing/diffusion when fluid moves over cells (no bulk motion) Mach 5 boost Bryan 95, Wadsley 08, Tasker & Bryan 08, Springel 10
15 Adaptive Mesh Refinement (AMR) CHALLENGE: POPULAR METHODS HAVE PROBLEMS Keplerian disk on a Cartesian (AMR) grid after 10 orbits Eulerian, well-studied, high-order Excessive mixing/diffusion when fluid moves over cells Geometric effects: - carbuncle instability (shocks) - loss of angular momentum - grid-alignment (disks) Disk torqued to align with grid Also beaten down with resolution, but expensive - Hahn 10: >>512 2 resolution to avoid grid-alignment Peery & Imlay 88, Mueller & Steinmetz 95, Hahn 10
16 Noh Implosion Sedov Explosion (noisy) (noisy) (grid breaks what should be spherical symmetry)
17 New Hybrid Lagrangian Godunov Methods (MFM/MFV/MMM)
18 Challenge: POPULAR METHODS FOR HYDRODYNAMICS HAVE PROBLEMS New Methods Combine (some) Advantages of Both Moving-meshes (AREPO), meshless finite-volume (GIZMO), high-order ALE methods Move with flow, no preferred geometry, but also accurate, high-order, and shock-capturing Less well-tested! AREPO: Springel 2010 TESS/DISCO: Duffel 2011 FVMHD3D: Gaburov 2012 GIZMO: Hopkins 2015 (arxiv: )
19 Challenge: POPULAR METHODS FOR HYDRODYNAMICS HAVE PROBLEMS Lanson & Vila 2008 Gaburov & Nitadori 2011 PFH 2014, 2015, 2016 Mesh-generating points move (if desired) x Volume is partitioned with a continuous kernel (MFM/MFV) or step function (moving-mesh)
20 Challenge: POPULAR METHODS FOR HYDRODYNAMICS HAVE PROBLEMS Lanson & Vila 2008 Gaburov & Nitadori 2011 PFH 2014, 2015, 2016 Integrate EOM over volume: equivalent to Reimann problem at effective face (quadrature)
21 New Methods Combine (some) Advantages of Both: GIZMO: disk after 100 orbits (BUT REMAIN LESS WELL-TESTED) Moving-meshes (AREPO), meshless finite-volume (GIZMO), high-order ALE methods Move with flow, no preferred geometry, but also accurate, high-order, and shock-capturing Grid noise is more severe AREPO: Springel 2010 TESS/DISCO: Duffel 2011 FVMHD3D: Gaburov 2012 GIZMO: Hopkins 2015 Rayleigh-Taylor sub-sonic turbulence
22 GIZMO: New Meshless Methods & Fluid Mixing ( Cartesian Grid Meshless Finite Volume Hopkins 2015 (arxiv: )
23 Summary SPH: Neighbor number: is it worth it? Hot halos: numerical mixing MHD: need even more neighbors Anisotropic diffusion: fundamental barriers Moving-mesh, meshless Godunov: Grid noise : Mach <~ 0.01 problems MHD: div-cleaning corrupts weak-field (va < 0.01*vturb,thermal) AMR: Angular momentum/grid alignment
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