relativistic smoothed particle hydrodynamics outline relativistic hydrodynamics relativistic sph...
TRANSCRIPT
Relativistic Smoothed Particle Hydrodynamics
Outline
• Relativistic hydrodynamics• Relativistic SPH• Entropy-based SPH• Shocks and artificial viscosity
C.E. Aguiar, T. KodamaU.F. Rio de Janeiro
T. Osada,Y. HamaU. São Paulo
Relativistic Hydrodynamics
gPuuT
)1,1,1,1(diag
),(
densitybaryon
densityenergy
pressure
densityenthalpy
g
u
n
P
P
v
unn
0 T
0 n
Energy-momentumconservation
Baryon-numberconservation
Baryon number conservation:
n
v
dt
d
vtdt
d
comoving derivative:
0)( un
0 T
dt
du
d
d
Pn
uwd
d
1)(
Energy-momentum conservation:
n/Pen/w enthalpy per baryon:
0
Pgnuu
n
t
Pw
dt
d
1)(
Energy equation:
)(1
vPdt
dE
n
PwE
Momentum equation:
Pdt
d
1qvq w
Entropy conservation:
0)(0
uTuv
entropy density (rest frame)
0dt
ds /n/s
v
dt
d
Lagrangian Equations
0dt
ds
)(1
vPdt
dE
Pdt
d
1q
v
dt
d
SPH
- L.Lucy, Astron.J. 82, 1013 (1977)- R.Gingold, J.Monaghan, MNRAS 181, 378 (1977)
• Developed to study gas dynamics in astrophysical systems. • Lagrangian method.• No grids.• Arbitrary geometries.• Equally applicable in 1, 2 and 3 space dimensions.
Reviews:- J. Monaghan, Annu. Rev. Astron. Astrophys. 30, 543 (1992)- L. Hernquist, N. Katz, Ap. J. Suppl. 70, 419 (1989)
Smoothing
xxxxxx dhWAAA S ),()()()(
)()()( 2hOAAS xx
h
x0 1),( xx dhW
)()()()()]()([ xxxxxx SSS BABABA
kernel smoothing),( hW x
Error:
Particles
b
bb
bbSPS W
AAA )(
)(
)()()( xx
x
xxx
xxxxx dWS )()()(
xxxxx
xx
dWAAS )()()(
)()(
b
bbSPS W )()()( xxxx
N
bbb
1
)()( xxx
"Monte-Carlo" sampling
b = baryon number of ''particle'' b
b
bb
bb W )(
)(
)()( xx
x
xvxv
b
bbb W )()()()( xxxvxvx
)()()()()]()([ xxxxxx SPSPSP BABABA
)()(0, xx AAhN SP
Different ways of writing SP estimates(we omit the SP subscript from now on):
b
bbb W )()()(
1)( xxxv
xxv
Derivatives
b
bb
bb W
AA )()( xxx
b
bb
bb W
AA )()( xxx
No need for finite differences and grids:
211 ii
i
AAA
i-1 i+1i
bab
b
bba W
vv
b
abab
bba W
vv)(
vvv )(
b
abaabba
a W)(1
)( vvv
More than one way of calculating derivatives:
AAA )(
b
baaabbaa WAAA )()()( xx
v
dt
d
b
abababaa W)()( vvv
)]()([ ttW bb
aba xx
bababab
a Wdt
d)( xx
aa
dt
dv
x
Moving the Particles
b
abab
bb
a
aab
a WPP
dt
dE22
vv
b
abab
b
a
ab
a WPP
dt
d22
q
22
1 PPP
dt
d q
aaaa w vq
2
1 vvv
PP)P(
dt
dE
a
aaaa
PwE
Energy equation
Momentum equation
)(
)()()(
)(
)()()(
00
0
aaa
a
aia
aa
ii
WE
ET
Wq
qT
xx
xxx
xx
xxx
aa
atotal
aa
atotal
EE
qP
Energy and Momentum
b
abbbb
abb
bba WsW
)()()( xxx s
0dt
dsa
Entropy equation
Particle Velocity
aaaaaa P,s,E,, vq ?
nPesnw /),(
1),(|| 2 snww qvq
1),/(|| 2 swq
),(,/, snPn v
snensnPsne )/(),(,),( 2
equation for
RSPH Equations
b
ababb
ba
a
ab
a WPP
dt
dEvv
22
b
abab
b
a
ab
a WPP
dt
d22
q
aa
dt
dv
x
b
abba W
1),/(|| 2 aaaaa swq
0dt
dsa
Baryon-Free Matter
TPn 0
0 T
0)( u
PuTd
d
1
0 Tu
dTdP
0
Pguu
vq T
PTE
v
dt
d
Pdt
d
1q
)(1
vPdt
dE
Lagrangian equations:
b
abab
b
a
ab
a WPP
sdt
d22
q
aa
dt
dv
x
b
abba Ws
aaaaaa
aa
aaa
n
n
/,/
1),( 2q
Entropy-based RSPH
b
ababb
ba
a
ab
a WPP
sdt
Edvv
22
b
abba W
Ultrarelativistic Pion Gas
42
30TP
32
15
2T
PPT 3
27766.02
)3(152
n
3
1
d
dPcs
- 8 - 6 - 4 - 2 0 2 4 6 8 1 0 1 2
x
0
0.2
0.4
0.6
0.8
1
1.2
entr
opy
dens
ity
exactSPH
N /L = 80h = 0.1dt = 0.05
R arefaction w aveP = (15/1282)1/3 4 /3
Pion Gas
Rarefaction Wave
0 2 4 6 8 10 12
x
0
0.1
0.2
0.3
0.4
0.5
entr
opy
dens
ity
exactSPH
Landau-Kalatn ikov so lutionP = (15/128 4 /3
N /L = 400h = 0.05dt = 0.02
Pion Gas
Landau Solution
Shock Waves
shock wave
x
numerical calculation
-50 -40 -30 -20 -10 0 10 20 30 40 50
x (fm )
0
1
2
3
4
5
6
entr
opy
dens
ity (
fm-3
)
N = 1000h = 0.5 fmdt = 0.25 fm /ctm ax = 50 fm /c
= 0 , = 0
Pion G as
Pion Gas
Shock Wave
Artificial Viscosity
gQPuuQPT )()(
v
dtdu
hfPQ
/
)(
sitybulk viscoQ
T
Qu
)(
u
Q
T
QQPu
Q
d
d)(
0,0
0,)()(
2hhhfTN
Thermodynamically normal matter:
0Q
Second Law of Thermodynamics:
Thermodynamically anomalous matter:
0,0
0,)()(
2hhhfTA
QPQ
E
vq
Q
T
Q
dt
d
v
T
QQP
dt
d
)(1
ET
QQP
dt
Ed
v)(1
b
abab
bb
a
aaba
aa WQPQP
ssdt
sd22
)( q
Dissipative RSPH
aa
dt
dv
x
b
abba Ws
aa
aaa
a
T
Qs
dt
sd
b
ababb
bba
a
aaba
aa WQPQP
ssdt
Esdvv
22
)(
b
abba W
-50 -40 -30 -20 -10 0 10 20 30 40 50
x (fm )
0
1
2
3
4
5
6
entr
opy
dens
ity (
fm-3
)
N = 1000h = 0.5 fmdt = 0.25 fm /ctm ax = 50 fm /c
= 2 , = 4
Pion G asShock Wave
Pion Gas
452shock
412shock
shock431
2
v1
1v9v
3
2/
/
/ )(
)(
Rankine - Hugoniot:
Pion Gas
1 2 3 4 5 6 7 8
ra tio o f entropy densities
0.5
0.6
0.7
0.8
0.9
1
shoc
k ve
loci
ty /
c
QGP + Pion Gas
c
c
TT,BraT
TT,aTP
4
4
322116302 /)/n(r,/a f
c
c
PP,BP
PP,P
43
3
)r(a
BT,
r
BP cc 11
4
Rarefaction Shock
QGP + Pions
-30 -20 -10 0 10 20 30 40 50
x (fm )
0
4
8
12
16
entr
opy
dens
ity (
fm-3
)
N = 3600h = 0.5 fmdt = 0.1 fm /ctm ax = 50 fm /c
= 4 , = 4
Q G P + P ion G as
Rarefaction Shock
QGP + Pions
-30 -20 -10 0 10 20 30 40 50
x (fm )
0
0.2
0.4
0.6
0.8
1
velo
city
/ c
N =3600h = 0.5 fmdt = 0.1 fm /ct = 50 fm /c= 4 , = 4
Q G P + P ion G as
Rarefaction Shock
QGP + Pions
-30 -20 -10 0 10 20 30 40 50
x (fm )
0
0.2
0.4
0.6
0.8
1
1.2
T /
TC
N = 3600h = 0.5 fmdt = 0.1 fm /ctm ax = 50 fm /c
= 4 , = 4
Q G P + P ion G as