SLIDE 1
Motivations
SPH perfect for NS-merger simulations
- No preferred geometry
- Resolution follows mass
- No need for background
density floor
National Science Foundation/LIGO/Sonoma State University/A. Simonnet
- Except….
- No GR
General Relativistic Smoothed Particle Hydrodynamics (GR-SPH) - - PowerPoint PPT Presentation
General Relativistic Smoothed Particle Hydrodynamics (GR-SPH) - - PowerPoint PPT Presentation
General Relativistic Smoothed Particle Hydrodynamics (GR-SPH) David Liptai Supervisors: Daniel Price and Paul Lasky Motivations Neutron star mergers! SPH perfect for NS-merger simulations - No preferred geometry - Resolution follows mass - No
SLIDE 2
SLIDE 3
Motivations
Tidal Disruption Events Tearing Discs and QPOs
Bonnerot et al. (2016)
Fake GR
☹
SLIDE 4
Motivations
Event Horizon Telescope Relativistic Pulsar Winds
Paredes-Fortuny et al. (2015) Credit: Hotaka Shiokawa
SLIDE 5
dρ∗ dt = −ρ∗ ∂vi ∂xi dpi dt = − 1 ρ∗ ∂(√−gP) ∂xi + √−g 2ρ∗ ✓ T µν ∂gµν ∂xi ◆ de dt = − 1 ρ∗ ∂(√−gPvi) ∂xi + −√−g 2ρ∗ ✓ T µν ∂gµν ∂t ◆
Equations of relativistic hydrodynamics
Continuity: Momentum: Energy:
| {z }
“Hydro”
| {z } | {z } | {z }
“GR”
SLIDE 6
Equations of relativistic hydrodynamics
Continuity: Momentum: Energy:
| {z }
“Hydro”
| {z } | {z }
| {z }
“GR” dρ∗
a
dt = 1 Ωa X
b
mb(vi
a − vi b)∂Wab(ha)
∂xi , dpa
i
dt = − X
b
mb √−gaPa Ωaρ∗2
a
∂Wab(ha) ∂xi + √−gbPb Ωbρ∗2
b
∂Wab(hb) ∂xi
- + f a
i ,
dea dt = − X
b
mb √−gaPavi
b
Ωaρ∗2
a
∂Wab(ha) ∂xi + √−gbPbvi
a
Ωbρ∗2
b
∂Wab(hb) ∂xi
- + Λa,
ρ∗
a =
X
b
mbWab(ha)
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Metrics and Coordinates
- Minkowski, Schwarzschild and Kerr
- Need in Cartesian-like coordinates
- A way to compute derivatives
- Choice of frame? (which observer?)
Checklist:
ds2 = − ✓ 1 − 2M r ◆ dt2 + dr2 1 − 2M
r
+ r2 dθ2 + sin2 θ dφ2 = − ✓ 1 − 2M r ◆ dt2 + ... dx2 + ... dx dy + ... dx dz + ...
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SLIDE 8
p
n+ 1
2
i
= pn
i + ∆t
2 f sph
i
(pn
i , xi,n),
p
m+ 1
2
i
= pm
i + ∆text
2 f ext
i
(p
m+ 1
2
i
, xi,m), xi,m+1 = xi,m + ∆text 2 dxi dt (p
m+ 1
2
i
, xi,m) +dxi dt (p
m+ 1
2
i
, xi,m+1)
- ,
pm+1
i
= p
m+ 1
2
i
+ ∆text 2 f ext
i
(p
m+ 1
2
i
, xi,m+1), pn+1
i
= p
n+ 1
2
i
+ ∆t 2 f sph
i
(pn+1
i
, xi,n+1)
(
)
Time Integration
- Preserve the Hamiltonian
properties of the system
- Operator splitting approach
- Time reversible (conserves
energy)
- Cost effective for 2nd order
Modified Leapfrog algorithm
Checklist:
SLIDE 9
Recovery of Primitive Variables
- Needs to be done after every
time-step
- Needs to rigorous and cheap
- Cannot solve explicitly
- Solve numerically with a
Newton-Raphson scheme
- Follow Tejeda (2012)
ρ∗ = √−gρ U 0, pi = U 0wgiµvµ, e = U 0 ⇥ wgiµvµvi − (1 + u)gµνvµvν⇤ , ρ = ??? vi = ??? u = ???
Checklist:
?
SLIDE 10
Tests: Schwarzschild metric
Radial Infall
vr(r) = 1 − 2M
r
q 1 − 2M
r0
s 2M ✓1 r − 1 r0 ◆
Ω = 1 r3/2
Circular orbit Precession
SLIDE 11
Liptai and Price 2018 (In prep.)
Tests: Kerr metric
Apsidal precession Retrograde Prograde
SLIDE 12
Epicyclic frequency
Tests: Kerr metric
Liptai and Price 2018 (In prep.)
Vertical-oscillation frequency
SLIDE 13
SLIDE 14
Nodal Precession
SLIDE 15
A mock tidal disruption event (TDE) in the Kerr metric
SLIDE 16
SLIDE 17
SLIDE 18
SLIDE 19
Tests: shock capturing
- 1D shock tubes
- Minkowski metric (special rel)
+ ✓dpi dt ◆
diss
+ ✓de dt ◆
diss
What should we use?
SLIDE 20
Chow & Monaghan (1997) Overly dissipative Siegler & Riffert (2000) No artificial conductivity
| {z } | {z }
Attempts at artificial dissipation in SR
SLIDE 21
Controlling artificial conductivity
✓dpa dt ◆
diss
⇠ X
b
mb ¯ ρab vsig ˆ rab · (pa pb) rW ab ✓dea dt ◆
diss
⇠ X
b
mb ¯ ρab vsig (ea eb) ˆ rab · rW ab
Non-relativistic
e = 1 2v2 + u ea − eb = 1 2αvisc
- v2
a − v2 b
- + αcond (ua − ub)
| {z }
Viscosity
| {z }
Conductivity
Relativistic
ea − eb = ... ??? ... e = v2 √ 1 − v2 (1 + u + P/ρ) + p 1 − v2(1 + u)
αvisc ω
- γav2
a − γbv2 b
- +
✓ 1 γa − 1 γb ◆ + αcond ua γa − ub γb
- |
{z }
| {z }
Viscosity Conductivity
SLIDE 22
Artificial viscosity only Artificial viscosity AND conductivity
| {z } | {z }
Liptai and Price 2018 (In prep.)
1D special relativistic shock tubes
SLIDE 23
1D special relativistic shock tubes
Ultra-relativistic 1D Sine wave perturbation 1D
| {z } | {z }
Liptai and Price 2018 (In prep.)
SLIDE 24 Publications of the Astronomical Society of Australia (PASA) c Astronomical Society of Australia 2017; published by Cambridge University Press. doi: 10.1017/pas.2017.xxx.
P h a n t
- m
: A s m
- t
h e d p a r t i c l e h y d r
- d
y n a m i c s a n d m a g n e t
- h
y d r
- d
y n a m i c s c
- d
e f
- r
a s t r
- p
h y s i c s
Daniel J. Price1∗, James Wurster2,1, Chris Nixon3, Terrence S. Tricco4,1, St´ even Toupin5, Alex Pettitt6, Conrad Chan1, Guillaume Laibe7, Simon Glover8, Clare Dobbs2, Rebecca Nealon1, David Liptai1, Hauke Worpel9,1, Cl´ ement Bonnerot10, Giovanni Dipierro11, Enrico Ragusa11, Christoph Federrath12, Roberto Iaconi13, Thomas Reichardt13, Duncan Forgan14, Mark Hutchison1, Thomas Constantino2, Ben Ayliffe15,1, Daniel Mentiplay1, Kieran Hirsh1 and Giuseppe Lodato11
1Monash Centre for Astrophysics (MoCA) and School of Physics and Astronomy, Monash University, Vic. 3800, Australia 2School of Physics, University of Exeter, Stocker Rd., Exeter EX4 4QL, UK 3Theoretical Astrophysics Group, Department of Physics & Astronomy, University of Leicester, Leicester LE1 7RH, UK 4Canadian Institute for Theoretical Astrophysics (CITA), University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada 5Institut d’Astronomie et d’Astrophysique (IAA), Universit´ e Libre de Bruxelles (ULB), CP226, Boulevard du Triomphe B1050 Brussels, Belgium 6Department of Cosmosciences, Hokkaido University, Sapporo 060-0810, Japan 7Univ Lyon, ENS de Lyon, CNRS, Centre de Recherche Astrophysique de Lyon UMR5574, F-69230, Saint-Genis-Laval, France 8Zentrum f¨ ur Astronomie der Universit¨ at Heidelberg, Institut f¨ ur Theoretische Astrophysik, Albert-Ueberle-Str 2, D-69120 Heidelberg, Germany 9AIP Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany 10Leiden Observatory, Leiden University, PO Box 9513, NL-2300 RA Leiden, the Netherlands 11Dipartimento di Fisica, Universit` a Degli Studi di Milano, Via Celoria 16, Milano, 20133, Italy 12Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia 13Department of Physics and Astronomy, Macquarie University, Sydney, Australia 14St Andrews Centre for Exoplanet Science and School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, UK 15Met Office, FitzRoy Road, Exeter, EX1 3PB, UK Abstract We present Phantom, a fast, parallel, modular and low-memory smoothed particle hydrodynamics and magnetohydrodynamics code developed over the last decade for astrophysical applications in three dimen-- sions. The code has been developed with a focus on stellar, galactic, planetary and high energy astrophysics
1 Introduction Numerical simulations are the ‘third pillar’ of astro- physics, standing alongside observations and analytic
- theory. Since it is difficult to perform laboratory ex-
periments in the relevant physical regimes and over the correct range of length and time-scales involved in most astrophysical problems, we turn instead to ‘numerical experiments’ in the computer for understanding and in-
- sight. As algorithms and simulation codes become ever
more sophisticated, the public availability of simulation
∗daniel.price@monash.educodes has become crucial to ensure that these experi- ments can be both verified and reproduced. Phantom is a smoothed particle hydrodynamics (SPH) code, written in Fortran 90, developed over the last decade. It has been used widely for studies of accre- tion (Lodato & Price, 2010; Nixon et al., 2012a; Rosotti et al., 2012; Nixon, 2012; Nixon et al., 2012b; Facchini et al., 2013; Nixon et al., 2013; Martin et al., 2014a,b; Nixon & Lubow, 2015; Coughlin & Nixon, 2015; For- gan et al., 2017) and turbulence (Kitsionas et al., 2009; Price & Federrath, 2010; Price et al., 2011; Price, 2012b; Tricco et al., 2016b) as well as for studies of the Galaxy 1
arXiv:1702.03930v1 [astro-ph.IM] 13 Feb 2017
3D Hydrodynamics
SLIDE 25
3D Special relativistic shocktubes
Mildly-relativistic 3D Ultra-relativistic 3D
| {z } | {z }
Liptai and Price 2018 (In prep.)
SLIDE 26
3D spherical blast wave
r vr 0.1 0.2 0.3 0.4 0.5 1 t=0.15
Maximum Lorentz factor ~6.4
SLIDE 27
Other 3D Hydro
Accretion disc around a Schwarzschild black hole
1×10-8 2×10-8 3×10-8 column density 1×10-9 2×10-9 3×10-9 column density t=1000 100 GM/c2
top-down side-on
Liptai and Price 2018 (In prep.)
density r 10 20
- 0.05
0.05 0.1 vr r 10 20 0.1 0.2 0.3 0.4 u r 10 20 0.1 0.2
Generalised spherically symmetric Bondi accretion (Schwarzschild)
x y
- 20
- 10
10 20
- 20
- 10
10 20 0.2 0.4 column density
SLIDE 28
Conclusions
- Orbital tests (Schwarzschild AND Kerr) are in excellent
agreement with theory
- We can handle relativistic shocks very well
- We have split artificial dissipation into viscosity and
conductivity
- Merged with PHANTOM to do full 3D-GRSPH simulations
SLIDE 29