2.3 Exact solution of the Riemann problem in SRHD

Let us first consider the one-dimensional special relativistic flow of a perfect fluid in the absence of a gravitational field. The Riemann problem then consists of computing the breakup of a discontinuity, which initially separates two arbitrary constant states L (left) and R (right) in the fluid (see Figure 1View Image with L ≡ 1 and R ≡ 5). For classical hydrodynamics the solution can be found, e.g., in [62]. In the case of SRHD, the Riemann problem was considered by Martí and Müller [180Jump To The Next Citation Point], who derived an exact solution for the case of pure normal flow generalizing previous results for zero initial velocities [276]. More recently, Pons, Martí and Müller [235Jump To The Next Citation Point] have obtained the general solution in the case of non-zero tangential speeds.

The solution to this problem is self-similar, because it only depends on the two constant states defining the discontinuity v L and v R, where v = (p,ρ,vx,vy, vz), and on the ratio (x − x )∕(t − t ) 0 0, where x0 and t0 are the initial location of the discontinuity and the time of breakup, respectively. Both in relativistic and classical hydrodynamics the discontinuity decays into two elementary nonlinear waves (shocks or rarefactions) which move in opposite directions towards the initial left and right states. Between these waves two new constant states vL∗ and vR∗ (note that vL∗ ≡ 3 and vR∗ ≡ 4 in Figure 1View Image) appear, which are separated from each other by a contact discontinuity moving with the fluid. Accordingly, the time evolution of a Riemann problem can be represented as

I → L ð’ē ← L ∗ 𝒞 R ∗ ð’ē → R, (14 )
where ð’ē and 𝒞 denote a simple wave (shock or rarefaction) and a contact discontinuity, respectively. The arrows (← / →) indicate the direction (left / right) from which fluid elements enter the corresponding wave.

As in the Newtonian case, the compressive character of shock waves (density and pressure rise across the shock) allows us to discriminate between shocks (ð’Ū) and rarefaction waves (ℛ):

{ ℛ ←(→ ) for pb ≤ pa, ð’ē ← (→) = ð’Ū for p > p , (15 ) ← (→ ) b a
where p is the pressure, and subscripts a and b denote quantities ahead and behind the wave. For the Riemann problem a ≡ L(R) and b ≡ L∗(R ∗) for ð’ē ← and ð’ē →, respectively. Thus, the possible types of decay of an initial discontinuity can be reduced to
∙ I → L ð’Ū ← L∗ 𝒞 R ∗ ð’Ū → R, pL < pL ∗ = pR ∗ > pR, ∙ I → L ð’Ū ← L∗ 𝒞 R ∗ ℛ → R, pL < pL ∗ = pR ∗ ≤ pR, (16 ) ∙ I → L R ← L ∗ 𝒞 R ∗ ℛ → R, pL ≥ pL ∗ = pR ∗ ≤ pR.

Across the contact discontinuity the density exhibits a jump, whereas pressure and normal velocity are continuous (see Figure 1View Image). As in the classical case, the self-similar character of the flow through rarefaction waves and the Rankine–Hugoniot conditions across shocks provide the relations to link the intermediate states vS∗ (S = L,R) with the corresponding initial states vS. They also allow one to express the normal fluid flow velocity in the intermediate states (vxS∗ for the case of an initial discontinuity normal to the x axis) as a function of the pressure pS∗ in these states.

The solution of the Riemann problem consists in finding the intermediate states L ∗ and R ∗, as well as the positions of the waves separating the four states (which only depend on L, L ∗, R ∗, and R). The functions ð’ē → and ð’ē ← allow one to determine the functions vxR∗(p) and vxL∗(p), respectively. The pressure p∗ and the flow velocity vx ∗ in the intermediate states are then given by the condition

x x x vR∗(p∗) = vL∗(p ∗) = v∗, (17 )
where p∗ = pL∗ = pR∗.

In the case of relativistic hydrodynamics, the major difference to classical hydrodynamics stems from the role of tangential velocities. While in the classical case the decay of the initial discontinuity does not depend on the tangential velocity (which is constant across shock waves and rarefactions), in relativistic calculations the components of the flow velocity are coupled by the presence of the Lorentz factor in the equations. In addition, the specific enthalpy also couples with the tangential velocities, which becomes important in the thermodynamically ultrarelativistic regime.

The functions x vS∗(p) are defined by

{ S vxS∗(p) = ℛ S(p), if p ≤ pS, (18 ) ð’Ū (p), if p > pS,
where S ℛ (p ) (S ð’Ū (p)) denotes the family of all states which can be connected by a rarefaction (shock) with a given state vS ahead of the wave.
View Image

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 p1 > p5, ρ1 > ρ5, and v1 = v2 = 0 separated by a diaphragm at xD. 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.

The fact that one Riemann invariant is constant across any rarefaction wave provides the relation needed to derive the function ℛS. In differential form, the function reads

dvx 1 1 ----= ± --------∘-----------------, (19 ) dp ρhW 2cs 1 + g(ξ±,vx, vt)
where t ∘------------- v = (vy)2 + (vz)2 is the absolute value of the tangential velocity, and
(vt)2(ξ2 − 1) g(ξ±,vx,vt) = -------±-x-2-, (20 ) (1 − ξ±v )
and where
∘ -------------------------------- vx(1 − c2) ± c (1 − v2)[1 − v2c2− (vx)2(1 − c2)] ξ± = --------s-----s-----------------s-------------s-, (21 ) 1 − v2c2s
the + (−) sign corresponding to S = R (S = L). In the previous expressions, c s stands for the local sound speed.

Considering that in a Riemann problem the state ahead of the rarefaction wave is known, the integration of Equation (19View Equation) allows one to connect the states ahead (S) and behind the rarefaction wave. Moreover, using Equation (21View Equation), the EOS, and the following relation obtained from the constraint hW vt = constant., that holds across the rarefaction wave,

( )1∕2 t t ----1 −-(vx-)2--- v = hSWSv S h2 + (h W vt)2 , (22 ) S S S
the ODE can be integrated, the solution being only a function of p. Let us point out that the integration of Equation (19View Equation) is along an adiabat of the EOS.

In the limit of zero tangential velocities, vt = 0, the function g does not contribute. In this limit and in the case of an ideal gas EOS one has

c c W 2dvx = ± --sdp = ± -sdρ, (23 ) γp ρ
(where γ is the adiabatic exponent of the EOS) recovering expression (30) in [180Jump To The Next Citation Point]. The equation can be then integrated to give [180Jump To The Next Citation Point]
ℛS (p) = (1 +-vS)A±-(p-) −-(1-−-vS), (24 ) (1 + vS)A± (p ) + (1 − vS)
( ) 2 √γ-−--1 − c(p) √ γ −-1 + csS ± √γ−1 A ±(p) = √-------------√------------ , (25 ) γ − 1 + c(p) γ − 1 − csS
the + (−) sign of A ± corresponding to S = L (S = R). In the above equation, csS is the sound speed of the state v sS, and c(p) is given by
( ) γ(γ − 1)p 1∕2 c(p) = (γ −-1)ρ-(p∕p-)1∕γ +-γp . (26 ) S S

The family of all states ð’ŪS(p), which can be connected through a shock with a given state vS ahead of the wave, is determined by the shock jump conditions. One obtains

( )⌊ ( ) ⌋−1 S ( x ----∘p −-pS-----)⌈ ( --1--- ----∘--vxS-------) ⌉ ð’Ū (p) = hSWSv S ± j(p) 1 − V (p)2 hSWS + (p − pS ) ρSWS ± j(p) 1 − V (p)2 , (27 ) ± ±
where the + (−) sign corresponds to S = R (S = L). V+(p) and V− (p) denote the shock velocities for shocks propagating to the right and left, respectively. They are given by
∘ -------------------------- ρ2SW 2SvxS ± j(p)2 1 + ρ2SW 2S(1 − (vxS)2)∕j(p)2 V±(p) = --------------ρ2-W-2+--j(p)2---------------. (28 ) S S
Tangential velocities in the initial states are hidden within the flow Lorentz factor WS. On the other hand, j (p ) stands for the modulus of the mass flux across the shock front,
┌ ------------------ ││ p − p j(p) = ││ --2----S--2-------, (29 ) ∘ hS −-h(p)-− 2hS- pS − p ρS
where the enthalpy h(p) of the state behind the shock can be obtained from the Taub adiabat,
(h h ) h2 − h2S = --+ -S- (p − pS). (30 ) ρ ρS
In the general case, the above nonlinear equation must be solved together with the EOS to obtain the post-shock enthalpy as a function of p. In the case of ideal gas EOS with constant adiabatic index, the post-shock density ρ can be easily eliminated, and the post-shock enthalpy is the (unique) positive root of the quadratic equation [180Jump To The Next Citation Point]
( ) (γ-−-1)(pS-−-p) 2 (γ-−-1)(pS-−-p) hS-(pS −-p-) 2 1 + γp h − γp h + ρ − h S = 0. (31 ) S

Finally, the tangential velocities in the post-shock states can be obtained from [235Jump To The Next Citation Point]

( 1 − (vx)2 )1∕2 vt = hSWSvtS -2----------t-2-. (32 ) h + (hSWSv S)

Figure 2View Image shows the solution of a particular mildly relativistic Riemann problem for different values of the tangential velocity. The crossing point of any two lines in the upper panel gives the pressure and the normal velocity in the intermediate states. The range of possible solutions in the (p, vx)-plane is marked by the shaded region. While the pressure in the intermediate state can take any value between pL and p R, the normal flow velocity can be arbitrarily close to zero in the case of an extremely relativistic tangential flow. The values of the tangential velocity in the states L ∗ and R ∗ are obtained from the value of the corresponding functions at x v in the lower panel of Figure 2View Image. The influence of initial left and right tangential velocities on the solution of a Riemann problem is enhanced in highly relativistic problems. We have computed the solution of one such problem (see Section 6.2.2 below, Problem 2) for different combinations of vt L and vt R. The initial data are p L = 103, ρL = 1, x vL = 0; pR = 10–2, ρR = 1, x vR = 0, and the 9 possible combinations of vtL,R = 0, 0.9, 0.99. The results are given in Figure 3View Image and Table 1, and a complete discussion can be found in [235Jump To The Next Citation Point].

View Image

Figure 2: Graphical solution in the (p,vx)-plane (upper panel) and in the (vt,vx)-plane (lower panel) of the relativistic Riemann problem with initial data pL = 1.0, ρL = 1.0, vxL = 0.0, and pR = 0.1, ρR = 0.125, vx R = 0.0 for different values of the tangential velocity vt = 0, 0.5, 0.9, 0.999, represented by solid, dashed, dashed-dotted and dotted lines, respectively. An ideal gas EOS with γ = 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 L∗ and R∗ is obtained from the value of the corresponding functions at vx in the lower panel, and I 0 gives the solution for vanishing tangential velocity. The range of possible solutions is given by the shaded region in the upper panel.
View Image

Figure 3: Analytical pressure, density and flow velocity profiles at t = 0.4 for the relativistic Riemann problem with initial data p L = 103, ρ L = 1.0, vx L = 0.0, and p R = 10–2, ρR = 1.0, x vR = 0.0, varying the values of the tangential velocities. From left to right, t vR = 0, 0.9, 0.99 and from top to bottom vtL =0, 0.9, 0.99. An ideal EOS with γ = 5/3 was assumed.

Table 1: Solution of the relativistic Riemann problem at t = 0.4 with initial data pL = 103, ρ L = 1.0, vx L = 0.0, p R = 10–2, ρ R = 1.0, and vx R = 0.0 for 9 different combinations of tangential velocities in the left (t vL) and right (t vR) initial state. An ideal EOS with γ = 5/3 was assumed. The various quantities in the table are: the density in the intermediate state left (ρL∗) and right (ρR∗) of the contact discontinuity, the pressure in the intermediate state (p∗), the flow speed in the intermediate state (vx ∗), the speed of the shock wave (Vs), and the velocities of the head (ξh) and tail (ξt) of the rarefaction wave.
t vL t vR ρL∗ ρR∗ p∗ x v∗ Vs ξh ξt
0.00 0.00 9.16 × 10–2 1.04 × 10+1 1.86 × 10+1 0.960 0.987 –0.816 +0.668
0.00 0.90 1.51 × 10 –1 1.46 × 10+1 4.28 × 10+1 0.913 0.973 –0.816 +0.379
0.00 0.99 2.89 × 10 –1 4.36 × 10+1 1.27 × 10+2 0.767 0.927 –0.816 –0.132
0.90 0.00 5.83 × 10–3 3.44 × 10+0 1.89 × 10–1 0.328 0.452 –0.525 +0.308
0.90 0.90 1.49 × 10–2 4.46 × 10+0 9.04 × 10–1 0.319 0.445 –0.525 +0.282
0.90 0.99 5.72 × 10–2 7.83 × 10+0 8.48 × 10+0 0.292 0.484 –0.525 +0.197
0.99 0.00 1.99 × 10–3 1.91 × 10+0 3.16 × 10–2 0.099 0.208 –0.196 +0.096
0.99 0.90 3.80 × 10–3 2.90 × 10+0 9.27 × 10–2 0.098 0.153 –0.196 +0.094
0.99 0.99 1.29 × 10–2 4.29 × 10+0 7.06 × 10–1 0.095 0.140 –0.196 +0.085

Finally, let us note that the procedure to obtain the pressure in the intermediate states p ∗ is valid for general EOS. Once p∗ has been obtained, the remaining state quantities and the complete Riemann solution,

( ) u = u x −-x0;uL, uR , (33 ) t − t0
can be easily derived. In Section 9.4, we provide two FORTRAN programs called RIEMANN (Section 9.4.1) and RIEMANN-VT (Section 9.4.2), which allow one to compute the exact solution of an arbitrary special relativistic Riemann problem for an ideal gas EOS with constant adiabatic index, both with zero and non-zero tangential speeds using the algorithm discussed above.

Solving a Riemann problem involves the solution of an algebraic equation for the pressure (Equation (17View Equation)). Moreover, the functional form of this equation depends on the wave pattern under consideration (see expressions (16View Equation). In a recent paper [241Jump To The Next Citation Point], Rezzolla and Zanotti have presented a procedure, suitable for implementation into an exact Riemann solver in one dimension, which removes the ambiguity arising from the wave pattern. That method exploits the fact that the expression for the relative velocity between the two initial states is a (monotonic) function of the unknown pressure, p∗, which determines the wave pattern. Hence, comparing the value of the (special relativistic) relative velocity between the initial left and right states with the values of the limiting relative velocities for the occurrence of the wave patterns (16View Equation), one can determine a priori which of the three wave patterns will actually result (see Figure 4View Image). In [242] the authors extend the above procedure to multi-dimensional flows.

View Image

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 p∗ and the filled squares indicate the limiting values for the relative velocities (&tidle;v12)2S, (&tidle;v12)SR, (&tidle;v12)2R (from [241]).

  Go to previous page Go up Go to next page