Schematic solution of a Riemann problem in special relativistic hydrodynamics. The initial state at
(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.
Graphical solution in the
-plane (upper panel) and in
the 
-plane (lower panel) of the relativistic Riemann problem
with initial data 
, 
, 
, and
, 
, 
for different values of the tangential
velocity 
,
represented by solid, dashed, dashed-dotted and dotted lines,
respectively. An ideal gas EOS
with 
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.
Analytical pressure, density and flow velocity profiles at
for the relativistic Riemann problem with initial data 
, 
, 
, and 
,
, 
,
varying the values of the tangential
velocities. From left to right, 
and
from top to bottom 
. An ideal EOS with 
was assumed.
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 [240]).
Schematic solution of the shock heating problem in spherical geometry. The initial state consists of a spherically symmetric flow of cold (
) 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
post-shock state is constant and at rest. The pre-shock
state, where the flow is self-similar, has a
density which varies as 
with
time 
and radius 
.
MPEG movie showing the evolution of the density distribution for the shock heating problem with an inflow velocity
in Cartesian coordinates. The reflecting wall
is located at 
. 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 (
). 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.
Generation and propagation of a relativistic blast wave (schematic). The large pressure jump at a discontinuity initially located at
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.
Density distribution for the relativistic blast wave Problem 1 defined in Table 7 at t=0.314 obtained with the code SPH-RS-gr (see Table 5) of Muir [204] using 5500 SPH particles and a 1D version of the code. The solid lines give the exact solutions.
Velocity distribution for the relativistic blast wave Problem 1 defined in Table 7 at t=0.314 obtained with the code SPH-RS-gr (see Table 5) of Muir [204] using 5500 SPH particles and a 1D version of the code. The solid lines give the exact solutions.
MPEG movie showing 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.
MPEG movie showing 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.
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.
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).
MPEG movie showing 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.
MPEG movie showing 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.
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.
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.
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
, respectively (from Scheck et
al. [255]). The black
lines are iso-contours of the beam
mass fraction with 
(outermost) and 
(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 MPEG movie
in Figure 19.
MPEG movie showing the logarithm of the density (normalized to the density of the ambient medium) for a hot leptonic relativistic jet at
y (from
Scheck et al. [255]).
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).
MPEG movie illustrating the propagation of a relativistic jet from a collapsar, whose progenitor is a rotating He star with a radius of
. 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 time-independent and spatially
localized energy deposition rate (
). One can
recognize the central spherical region (black circle) of
radius 
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.)