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

qq

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