Figure 1:
Schematic solution of a Riemann problem in special relativistic hydrodynamics. The initial state at t = 0 (top figure) consists of two constant states 1 and 5 with , , and separated by a diaphragm at . The evolution of the flow pattern once the diaphragm is removed (middle figure) is illustrated in a spacetime diagram (bottom figure) with a shock wave (solid line) and a contact discontinuity (dashed line) moving to the right. The bundle of solid lines represents a rarefaction wave propagating to the left. 

Figure 2:
Graphical solution in the plane (upper panel) and in the plane (lower panel) of the relativistic Riemann problem with initial data = 1.0, = 1.0, = 0.0, and = 0.1, = 0.125, = 0.0 for different values of the tangential velocity = 0, 0.5, 0.9, 0.999, represented by solid, dashed, dasheddotted and dotted lines, respectively. An ideal gas EOS with = 1.4 was assumed. The crossing point of any two lines in the upper panel gives the pressure and the normal velocity in the intermediate states. The value of the tangential velocity in the states and is obtained from the value of the corresponding functions at in the lower panel, and gives the solution for vanishing tangential velocity. The range of possible solutions is given by the shaded region in the upper panel. 

Figure 3:
Analytical pressure, density and flow velocity profiles at t = 0.4 for the relativistic Riemann problem with initial data = 10^{3}, = 1.0, = 0.0, and = 10^{–2}, = 1.0, = 0.0, varying the values of the tangential velocities. From left to right, = 0, 0.9, 0.99 and from top to bottom =0, 0.9, 0.99. An ideal EOS with = 5/3 was assumed. 

Figure 4:
Relative velocity between the two initial states 1 and 2 as a function of the pressure at the contact discontinuity. Note that the curve shown is given by the continuous joining of three different curves describing the relative velocity corresponding to two shocks (dashed line), one shock and one rarefaction wave (dotted line), and two rarefaction waves (continuous line), respectively. The joining of the curves is indicated by filled circles. The small inset on the right shows a magnification for a smaller range of and the filled squares indicate the limiting values for the relative velocities , , (from [241]). 

Figure 5:
Schematic solution of the shock heating problem in spherical geometry. The initial state consists of a spherically symmetric flow of cold (p = 0) gas of unit rest mass density having a highly relativistic inflow velocity everywhere. A shock is generated at the center of the sphere, which propagates upstream with constant speed. The postshock state is constant and at rest. The preshock state, where the flow is selfsimilar, has a density which varies as = (1 + t/r)^{2} with time t and radius r. 

Figure 6:
Movie The evolution of the density distribution for the shock heating problem with an inflow velocity v_{1} = –0.99999 c in Cartesian coordinates. The reflecting wall is located at x = 0. The adiabatic index of the gas is 4/3. For numerical reasons, the specific internal energy of the inflowing cold gas is set to a small finite value ( = 10^{–7} W_{1}). The final frame of the movie also shows the analytical solution (blue lines). The simulation has been performed on an equidistant grid of 100 zones. 

Figure 7:
Generation and propagation of a relativistic blast wave (schematic). The large pressure jump at a discontinuity initially located at r = 0.5 gives rise to a blast wave and a dense shell of material propagating at relativistic speeds. For appropriate initial conditions both the speed of the leading shock front and the velocity of the shell approach the speed of light, producing very narrow structures. 

Figure 8:
Density distribution for the relativistic blast wave Problem 1 defined in Table 7 at t = 0.314 obtained with the code SPHRSgr (see Table 5) of Muir [204] using 5500 SPH particles and a 1D version of the code. The solid lines give the exact solutions. 

Figure 9:
Velocity distribution for the relativistic blast wave Problem 1 defined in Table 7 at t = 0.314 obtained with the code SPHRSgr (see Table 5) of Muir [204] using 5500 SPH particles and a 1D version of the code. The solid lines give the exact solutions. 

Figure 10:
Movie The evolution of the density distribution for the relativistic blast wave Problem 1 defined in Table 7. The final frame of the movie also shows the analytical solution (blue lines). The simulation has been performed with relativistic PPM on an equidistant grid of 400 zones. 

Figure 11:
Movie The evolution of the density distribution for the relativistic blast wave Problem 2 defined in Table 7. The final frame of the movie also shows the analytical solution (blue lines). The simulation has been performed with relativistic PPM on an equidistant grid of 2000 zones. 

Figure 12:
Results from [295] for the relativistic blast wave Problems 1 (left column) and Problem 2 (right column), respectively. Relativistic Glimm’s method is only used in regions with steep gradients. Standard finite difference schemes are applied in the smooth remaining part of the computational domain. In the above plots, Lax and LW stand for Lax and Lax–Wendroff methods, respectively; G refers to pure Glimm’s method. 

Figure 13:
The top panel shows a sequence of snapshots of the density profile for the colliding relativistic blast wave problem up to the moment when the waves begin to interact. The density profile of the new states produced by the interaction of the two waves is shown in the bottom panel (note the change in scale on both axes with respect to the top panel). 

Figure 14:
Movie The evolution of the density distribution for the colliding relativistic blast wave problem up to the interaction of the waves. The final frame of the movie also shows the analytical solution (blue lines). The computation has been performed with relativistic PPM on an equidistant grid of 4000 zones. 

Figure 15:
Movie The evolution of the density distribution for the colliding relativistic blast wave problem around the time of interaction of the waves at an enlarged spatial scale. The final frame of the movie also shows the analytical solution (blue lines). The computation has been performed with relativistic PPM on an equidistant grid of 4000 zones. 

Figure 16:
Time evolution of a light, relativistic (beam flow velocity equal to 0.99) jet with large internal energy. The logarithm of the proper rest mass density is plotted in grey scale, the maximum value corresponding to white and the minimum to black. 

Figure 17:
Logarithm of the proper rest mass density and energy density (from top to bottom) of an evolved, powerful jet propagating through the intergalactic medium. The white contour encompasses the jet material responsible for the synchrotron emission. 

Figure 18:
Snapshots of the logarithm of the density (normalized to the density of the ambient medium) for a cold baryonic (top panel), a cold leptonic (central panel) and a hot leptonic (bottom panel) relativistic jet at t 6.3 × 10^{6} y, respectively (from Scheck et al. [256]). The black lines are isocontours of the beam mass fraction with X = 0.1 (outermost) and X = 0.9 (innermost). These values correspond to the boundaries of the cocoon and the beam, respectively. The time evolution of the hot leptonic model is shown in the movie in Figure 19. 

Figure 19:
Movie The logarithm of the density (normalized to the density of the ambient medium) for a hot leptonic relativistic jet at t 6.3 × 10^{6} y (from Scheck et al. [256]). 

Figure 20:
Computed radio maps of a compact relativistic jet showing the evolution of a superluminal component (from left to right). Two resolutions are shown: present VLBI resolution (white contours) and resolution provided by the simulation (black/white images). 

Figure 21:
Movie The propagation of a relativistic jet from a collapsar, whose progenitor is a rotating He star with a radius of 3 × 10^{10} cm. All three panels show the rest mass density distribution. The left panel displays the computational domain up to the head of the jet (note the changing axis and color scales). The upper right panel shows the central region (scale fixed) where the jet forms due to a prescribed timeindependent and spatially localized energy deposition rate (10^{50} erg s^{–1}). One can recognize the central spherical region (black circle) of radius 2 × 10^{7} cm which was excised from the computational domain. It contains a (rotating) black hole of initially three solar masses accreting matter through the inner grid boundary. The lower right panel provides a global view (scale fixed) of the computational domain up to the surface of the He star progenitor. (Movie courtesy of M.A. Aloy.) 
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