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Progress of Theoretical Physics Supplement No. 138, 2000
Applications of Artificial Wind Numerical Scheme for Relativistic Hydrodynamics in Astrophysics Hui-Min Zhang,∗) Igor V. Sokolov, Kyoko Furusawa and Jun-Ichi Sakai Laboratory for Plasma Astrophysics, Faculty of Engineering, Toyama University Toyama 930-8555, Japan (Received October 11, 1999) We apply a new way for the construction of efficient non-oscillation shock-capturing schemes, namely artificial wind (AW) scheme, for relativistic hydrodynamics (RHD) simulations in astrophysics. We also discuss physical foundations for computational RHD. An appropriate choice for a realistic equation of state of the matter at relativistic temperatures drastically facilitates the procedure for numerical integration of hydrodynamics equations. The time-consuming iteration approach to solve the coupling equation is avoided. We have developed a 2-D relativistic hydrodynamics code based on the AW scheme and the new RHD. Here we present two applications of the code in astrophysics, namely, numerical simulations of ultra-relativistic jet propagation and relativistic Richtmyer-Meshkov instability.
Due to numerical difficulties arising from strong relativistic shocks, motions with large value of the Lorentz factor as well as due to the complicated structure of the relativistic equations, relativistic fluid simulations are traditionally believed to be much more difficult as compared with usual hydrodynamic simulations. Furthermore, the coupling equation between the primitive variables and the conservative ones through Lorentz factor γ is a quartic equation which should be solved at each cell several times of iteration using Newton Raphson procedure to update the numerical solution for one time step. As a result, the relativistic codes are drastically time-consuming. Here, we apply a new way for the construction of efficient non-oscillation shockcapturing scheme, i.e., artificial wind (AW) scheme, for relativistic hydrodynamics simulation. The basic idea of AW scheme is to solve the governing equations in different steadily moving frames of reference chosen in such a way that the flow is supersonic there, resulting in simple upwind formulas. The concept and principle of the AW scheme is given in an accompanying paper. 1) Recently, we proposed a new form of computational relativistic hydrodynamics (RHD). The key point is a special equation of state (EOS) for the matter at relativistic temperature. 2) The new EOS is much more realistic, and can drastically facilitate the procedure for explicitly expressing the conservative variables via the primitive ones, and reversely, expressing the primitive variables via the conservative ones. In fact, in the new form of RHD, conservative variables are related to primitive variables through a simple second-order algebraic equation, the time-consuming iteration to solve the coupling equation is avoided. Here, we present two astrophysical applications. The first one is ultra-relativistic jet propagating through a homogeneous medium. The simulation has been run on ∗)
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Applications of Artificial Wind Numerical Scheme
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800 × 1600 active zones with high spatial resolutions 40 zones per beam radius Rb . Initially, the beam velocity is vb = 0.995c (Lorentz γ = 10), the beam Mach number is Mb = 15, and the ratio of the rest-mass density of the beam to the ambient medium is η = ρb /ρm = 0.01. Figure 1 displays the proper rest-mass density on a logarithmic scale. Simulation with ≤ 1.3 × 106 computational cells requires RAM less than 52 MB and ≈ 2 × 10−6 CPU seconds per zones and time step (on a Windows NT work station with 256 MB memory and CPU frequency 275 MHz). 10
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Fig. 1. Relativistic jet propagation. The grey scale shows density distribution. 80 60 40
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Fig. 2. Relativistic Richtmyer-Meshkov instability. The density (gray) and velocity (arrows ) distributions at the early (left panel) and the late (right panel) stage. The velocity shown in the right panel is relative velocity δv = v − vc , where vc is the propagation velocity of the interface.
One more application is relativistic Richtmyer-Meshkov (RM) instability. 3), 4) The initial data of three areas (from the left to the right) are ρL : ρM : ρR = 1 : 0.5 : 50, vL = 0.90c, vM = vR = 0, and pL = pM = pR = 0.01. One can see when a shock wave encounters a fluid discontinuity, small perturbations at the interface can grow into nonlinear structures having the form of “bubbles” and vortical flow. Thus we demonstrate that the AW scheme and the new form of RHD provides us with a way toward easy and efficient computational RHD. References 1) 2) 3) 4)
I. V. Sokolov, H.-M. Zhang and J. I. Sakai, this issue. H.-M. Zhang, I. V. Sokolov and J. I. Sakai, submitted to Astrophys. J. R. D. Richtmyer, Comm. Pure & Appl. Math. 13 (1960), 297. E. E. Meshkov, NASA Tech. Trans. F-13 (1970), 074.
