The new computer program for three dimensional relativistic hydrodynamical model

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The new computer program for three dimensional
relativistic hydrodynamical model

• Daniel Kikola
• Marcin Slodkowski

Heavy Ion Reaction Group Faculty of Physics
Warsaw University of Technology
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Contents

• Introduction
• Main concepts
• Algorithms for numerical relativistic
  hydrodynamics
• Program structure
• Tests results
• Plans for future

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Fluid Dynamics for Relativistic Nuclear Collisions

• Landau 1953
• first attempt to describe hadron-hadron
  collisions by fluid dynamics
• The Landau model assumes that the slab has no
  initial collective velocity and that rapid
  thermalization takes place which is completed at
  t 0
• equation of state has the simple
  ultrarelativistic form p c2se, c2s const.

• The Bjorken Model
• collective velocity of matter is of the scaling
  form v z/t

Fluid dynamics

• Idea
• treat nuclear matter as continuous medium
• describe evolution of the system by equations of
  motions of ideal fluid dynamics

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Hydrodynamic model

• very easy way to describe evolution of nuclear
  matter

• assumption matter is in local
  thermodynamical equilibrium

• Ingredients
• input
• equation of state
• initial conditions
• relativistic ideal hydrodynamics equations
• freeze out

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Main concept of fluid dynamics
After the first stage of ions collision, a state
of matter is represented by fluid dynamic.
Evolution of fluid dynamics is simulated using
finite difference methods.
Resolving differential equations on 3D nets of
physical cells represents time-space evolution
of matter
- local energy-momentum conservation
- local charge conservation
- energy density in the local rest frame of
fluid p - pressure in the local rest frame of
fluid n - charge density in the local rest
frame of fluid v - fluid 3- velocity
where
-the metric tensor
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Time-space system evolution
Define
R -net charge density in calculational frame
(laboratory frame), E - energy density in
calculational frame, M - momentum density in
calculational frame, - energy density in the
local rest frame of fluid p - pressure in the
local rest frame of fluid n - charge density in
the local rest frame of fluid
Equation of state (EoS)
if gtgt n
transformation from the calculational frame to
the local rest frame of the fluid
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Algorithms for numerical relativistic
hydrodynamics

• Task solving equations of type
• (hyperbolic conservation law )

• Way finite differences scheme

• Numerical algorithm can be constructed simply by
  solving a sequence of Riemann problems for the
  discontinuities at all cell boundaries in a given
  time step.

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The RHLLE Flux (Relativistic HartenLaxvan
LeerEinfeldt )

• Godunov-type algorithm
• not employ the full solution of the Riemann
  problem but approximate it by a region of
  constant flow between UL and UR

• bL lt 0 and bR gt 0 are the so-called signal
  velocities
• they characterize the velocities which inform
  about the decay of the discontinuity

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HLLE algorithm

• Density ULR
• is determined by integrating hyperbolic
  conservation law over a fixed interval xmin,
  xmax, xminlt bLt, xmax gt bRt.

• Flux F(ULR)
• is determined by integrating hyperbolic
  conservation law over the fixed interval 0,
  xmax or xmin, 0

• Estimation for the signal velocities
• the relativistic addition (subtraction) of flow
  velocities and sound velocities in the respective
  cells adjacent to the cell boundary

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MUSTA FORCE Flux

• MUSTA (MUlti STAge) approach develops upwind
  numerical fluxes by utilizing centred fluxes in a
  multi-stage predictor-corrector fashion.

• Effectively, MUSTA can be regarded as an
  approximate Riemann solver in which the predictor
  step opens the Riemann fan and the corrector step
  makes use of the information extracted from the
  opened Riemann fan

• FORCE centred flux used

• The key idea to open the Riemann fan by solving
  the local Riemann problem by evolving in time the
  initial data UL, UR via the governing equations
  and does not explicitly make use of wave
  propagation information in the construction of
  the numerical flux

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The procedure to evaluate the MUSTA-FORCE Flux

• The multi-staging (or local time stepping) is
  started by setting

If the prescribed number of stages K has been
reached, then STOP
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Accuracy improvement

• Second order accuracy in space - MUSCL procedure
• piecewise linear approximation in each cell

• Second order accuracy in time
• half step update method
• Runge-Kutta second order method

• Solution in three space dimensional - operator
  splitting method
• the full 3-dimensional solution is constructed to
  solve sequentially three one-dimensional problems
• to minimize systematical errors - random
  permutation of order of integration over x,y,z in
  each cell

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Hydrodynamical computer program as hybrid model

• Hydrodynamical computer program is able to
  co-operate with other kinetic model to describe
  evolution of nuclear matter
• Hydro model works as input for kinetic model
• Hydrodynamical computer program and other
  kinetic program may form hydrokinetic approach to
  heavy ion collisions

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Algorithm block scheme
Main space nets cells loops Kx,Ky,Kz
If occur freeze-out jump to calculation momenta
spectra of particles modules
Main time nets cells loop Tmax
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Code scheme
Main module main.c
io.h module
read_parameters(...) read_data(...)
calculate(...) write_data(...)
hydro.h module
fluid(...) hadronization(...)
fluid_dynamic.h module
MustaForce.h module
MustaForce(...) MustaForceWithRungeKutta(...)
hydro_init(...) integrating(...)
evpn_calculating(...) V_calc(...)
Conditions of compilation
hadronization.h module
hlle.h module
particles(...) particle_momenta(...)
IsCellFreezeOut(...) FreezeOutHyperSurface_init(.
..) FreezeOutHyperSurfaceclean(...)
HLLEfirstOrder(...) HLLEsecondOrder(...)
HLLEsecondOrderWithRungeKutt(...)
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Parallel network solution for solving hydro
equations more effectively
Scale able network computers solution for
hydrogrid process
x --gt (0 50) y --gt (0 50) z --gt (0 50)
Hydrogrid process
Parallel hydrogrid's simulation for nets cells
x --gt (50 100) y --gt (0 50) z --gt (0 50)
TCP/IP
Hydroserver process
Hydrogrid process
...
x --gt (50 100) y --gt (50 100) z --gt (50 100)
Hydrogrid process
hydro server synchronization for time step in
whole simulation
Matrix of time step
32 32 32 32 32 32 32 32 32 32 32 32 32 32 32 32
Each hydrogrid process in the same level of time
evolution
space cube divided by small cube simulated in
hydrogrids
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Test results

• Shock tube problem
• Shock tube problem is standard test for
  hydrodynamic code
• solution consist of rerefaction wave, contact and
  shock wave.
• Exact time-dependent solution is known and can be
  compared with the solution computed applying
  numerical discretizations.
• the initial conditions of the shock-tube problem
  are composed by two uniform states separated by a
  discontinuity its special case of well known
  Riemann Problem.

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Shock tube problem I
Test results
?x 0.02 ?t ?x/4 grid of 500 zones Number of
time steps 800 EoS ideal gas with an adiabatic
index ? 1.4
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Shock tube problem II
Test results
?x 0.02 ?t ?x/4 grid of 500 zones Number of
time steps 800 EoS ideal gas with an adiabatic
index ? 1.4
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Test results
Number of cells along z-direction  K max Z
400, grid spacing dz   0.02 time step dt
0.005 Number of time steps Tmax 400 EoS p
1/3
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Test results
Number of cells along x-direction  K maxX
400, Number of cells along y-direction KmaxY
400, grid spacing dx dy 0.025 time step dt
0.005 Number of time steps Tmax 600 EoS p
1/3
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Test results
2D Bjorken-like expansion with transverse
cylindrically symmetric flows
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Plans for future

• Accelerate computer simulation by adaptation
  computers parallel network solution PVM (Parallel
  Virtual Machine)
• Develop hadronization module (The kinetic
  definition of energy-momentum nets of cells)
• Phase 1 adaptation Cooper-Frye formula
• Phase 2 adaptation continuum emission proposed
  by prof. Y. Siniukov
• Parallel simulation of space-time system
  evolution and calculation of probability particle
  detection. Continuum freeze-out process for
  emission particle from area of simulation

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Team leaders
With co-operation with
dr. Wiktor Peryt
prof. Jan Pluta
prof. Yuri Sinukov Bogolyubov Institute for
Theoretical Physics Kiev
Team members
Marcin Slodkowski
Daniel Kikola
Marek Szuba
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References
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Thank you for your attention
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Bjorken expansion test
Number of cells along z-direction  K max Z
400, grid spacing dz   0.02 time step dt
0.005 Number of time steps Tmax 600 Cs2
const 1/3 t0 5 fm, t 7 fm
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2D Bjorken-like expansion with transverse
cylindrically symmetric flows
Number of cells along x-direction KmaxX
400, Number of cells along y-direction KmaxY
400, grid spacing dx dy 0.025 time step dt
0.005 Number of time steps Tmax 600
Cs2 const 1/3 t0 5 fm, t 8 fm R0 t0 -1.0