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The new computer program for three dimensional relativistic hydrodynamical model

The new computer program for three dimensional relativistic hydrodynamical model. Daniel Kikoła Marcin Słodkowski. Heavy Ion Reaction Group Faculty of Physics Warsaw University of Technology. C ontent s. Introduction Main concepts Algorithms for numerical relativistic hydrodynamics

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The new computer program for three dimensional relativistic hydrodynamical model

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  1. The new computer program for three dimensional relativistic hydrodynamical model Daniel Kikoła Marcin Słodkowski Heavy Ion Reaction Group Faculty of Physics Warsaw University of Technology International School on Quark-Gluon Plasma, Torino, Italy 2005

  2. Contents • Introduction • Main concepts • Algorithms for numerical relativistic hydrodynamics • Program structure • Tests results • Plans for future International School on Quark-Gluon Plasma, Torino, Italy 2005

  3. 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 International School on Quark-Gluon Plasma, Torino, Italy 2005

  4. 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” International School on Quark-Gluon Plasma, Torino, Italy 2005

  5. 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 International School on Quark-Gluon Plasma, Torino, Italy 2005

  6. 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 >> n transformation from the calculational frame to the local rest frame of the fluid International School on Quark-Gluon Plasma, Torino, Italy 2005

  7. - intercell numerical fluxes. 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. International School on Quark-Gluon Plasma, Torino, Italy 2005

  8. The RHLLE Flux (Relativistic Harten–Lax–van Leer–Einfeldt ) • 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 < 0 and bR > 0 are the so-called signal velocities • they characterize the velocities which inform about the decay of the discontinuity International School on Quark-Gluon Plasma, Torino, Italy 2005

  9. HLLE algorithm • Density ULR • is determined by integrating hyperbolic conservation law over a fixed interval [xmin, xmax], xmin< bLt, xmax > bRt. • Flux F(ULR) • is determined by integrating hyperbolic conservation lawover 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 International School on Quark-Gluon Plasma, Torino, Italy 2005

  10. 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 International School on Quark-Gluon Plasma, Torino, Italy 2005

  11. Step 1: Flux evaluation Step 2: Opening of Riemann fan Step 3: Goto step 1 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 International School on Quark-Gluon Plasma, Torino, Italy 2005

  12. 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 International School on Quark-Gluon Plasma, Torino, Italy 2005

  13. Hydrodynamical computer program as hybrid model • Hydrodynamical computer program is able to co-operatewith 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 International School on Quark-Gluon Plasma, Torino, Italy 2005

  14. 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 International School on Quark-Gluon Plasma, Torino, Italy 2005

  15. 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(...) International School on Quark-Gluon Plasma, Torino, Italy 2005

  16. Parallel network solution for solving hydro equations more effectively Scale able network computers solution for hydrogrid process x --> (0 : 50) y --> (0 : 50) z --> (0 : 50) Hydrogrid process Parallel hydrogrid's simulation for nets cells TCP/IP x --> (50 : 100) y --> (0 : 50) z --> (0 : 50) Hydroserver process Hydrogrid process ... x --> (50 : 100) y --> (50 : 100) z --> (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 International School on Quark-Gluon Plasma, Torino, Italy 2005

  17. 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. International School on Quark-Gluon Plasma, Torino, Italy 2005

  18. Test results Shock tube problem I ∆x = 0.02 ∆t = ∆x/4 grid of 500 zones Number of time steps: 800 EoS: ideal gas with an adiabatic index =1.4 International School on Quark-Gluon Plasma, Torino, Italy 2005

  19. Test results Shock tube problem II ∆x = 0.02 ∆t = ∆x/4 grid of 500 zones Number of time steps: 800 EoS: ideal gas with an adiabatic index =1.4 International School on Quark-Gluon Plasma, Torino, Italy 2005

  20. Comparison with Bjorken model 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 International School on Quark-Gluon Plasma, Torino, Italy 2005

  21. 2D Bjorken-like expansion with transverse cylindrically symmetric flows 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 Test results International School on Quark-Gluon Plasma, Torino, Italy 2005

  22. Test results 2D Bjorken-like expansion with transverse cylindrically symmetric flows International School on Quark-Gluon Plasma, Torino, Italy 2005

  23. 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 International School on Quark-Gluon Plasma, Torino, Italy 2005

  24. Team leaders: Withco-operation with dr. Wiktor Peryt prof. Yuri Sinukov Bogolyubov Institute for Theoretical Physics Kiev prof. Jan Pluta Team members: Marcin Słodkowski Daniel Kikoła Marek Szuba International School on Quark-Gluon Plasma, Torino, Italy 2005

  25. D.H. Rischke, S. Bernard, J.A. Maruhn "Relativistic Hydrodynamics for Heavy-Ion Collisions - I. General Aspects and Expansion into Vacuum", nucl-th/9504018 • S. Bernard, J.A. Maruhn, W. Greiner, D.H. Rischke, "Relativistic Hydrodynamics for Heavy-Ion Collisions: Freeze-Out and Particle Spectra", nucl-th/9602011 • D.H. Rischke, "Fluid Dynamics for Relativistic Nuclear Collisions", nucl-th/9809044 • H. Miao, Z. Ma, Ch. Gao, "Hydrodynamic Evolution of Spherical Fireball In Relativistic Heavy Ion Collisions", hep-ph/0303134 • W. N. Zhang, M. J. Efaaf, Cheuk-Yin Wong, M. Khaliliasr, "Pion Interferometry for Hydrodynamical Expanding Source with a Finite Baryon Density", nucl-th/0404047 • O. Socolowski Jr, F. Grassi, Y. Hama, T. Kodama, "Fluctuations of the Initial Conditions and the Continuous Emission in Hydro Description of Two-Pion Interferometry", hep-ph/0405181 • Yu.M. Sinyukov, S.V. Akkelin and Y.Hama, "On freeze-out problem in hydro-kinetic approach to A+A collisions", nucl-th/0201015 • E F Toro, "Multi-Stage Predictor-Corrector Fluxes for Hyperbolic Equations", NI03037-NPA References International School on Quark-Gluon Plasma, Torino, Italy 2005

  26. Thank you for your attention International School on Quark-Gluon Plasma, Torino, Italy 2005

  27. 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 International School on Quark-Gluon Plasma, Torino, Italy 2005

  28. 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 International School on Quark-Gluon Plasma, Torino, Italy 2005

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