Hybrid model with viscous relativistic hydrodynamics: a role of constraints on the shear-stress tensor

Abstract

We present the hybrid model connecting the parton–hadron–string dynamic model (PHSD) and a hydrodynamic model taking into account shear viscosity within the Israel–Stewart approach. The numerical scheme, initialization, and particlization procedure are discussed in detail. The performance of the code is tested on the pion and proton rapidity and transverse mass distributions calculated for Au + Au and Pb+Pb collisions at AGS–SPS energies. The influence of the switch time from transport to hydro models, the viscous parameter, and freeze-out time are discussed. Since the applicability of the Israel–Stewart hydrodynamics assumes the perturbative character of the viscous stress tensor, $\pi ^{\mu \nu }$, which should not exceed the ideal energy–momentum tensor, $T_{\mathrm{id}}^{\mu \nu }$, hydrodynamical codes usually rescale the shear stress tensor if the inequality $\Vert \pi ^{\mu \nu }\Vert \ll \Vert T_{\mathrm{id}}^{\mu \nu }\Vert$ is not fulfilled in some sense. There are several conditions used in the literature and we analyze in detail the influence of different conditions and values of the cut-off parameter on observables. We show that the form of the corresponding condition plays an important role in the sensitivity of hydrodynamic calculations to the viscous parameter – a ratio of the shear viscosity to the entropy density, $\eta /s$. It is shown that the constraints used in the vHLLE and MUSIC models give the same results for the observables. With these constraints, the rapidity distributions and transverse momentum spectra are most sensitive to a change of the $\eta /s$ ratio. We demonstrate that these constraints do not guarantee that each element of the $\pi ^{\mu \nu }$ tensor is smaller than the corresponding element $T_{\mathrm{id}}^{\mu \nu }$. As an alternative, a strict condition is used. When applied it reduces the sensitivity of the proton and pion momentum distributions to the viscosity parameter. We performed global fits of the rapidity and transverse mass distributions of pion and protons. It was also found that $\eta /s$ as a function of the collision energy monotonically increases from $E_{\mathrm{lab}}=6\,A\text {GeV}$ up to $E_{\mathrm{lab}}=40\,A\text {GeV}$ and saturates for higher SPS energies. We observe that it is difficult to reproduce simultaneously pion and proton rapidity distribution within our model with the present choice of the equation of state without a phase transition.

Appendices

Appendix A: Numerical realization

In this Appendix we discuss the numerical scheme used to integrate the hydrodynamic equations (1b), (9), and (6). The 10-dimensional vector $\mathbf {S}$ in the right-hand side of Eq. (10) can be written as a combinations of two 5-dimensional vectors

$$\begin{aligned} \mathbf {S}=\big (\mathbf {S}_{\mathrm{cons}},\mathbf {S}_\pi \big ) \end{aligned}$$

(A1)

corresponding to the conservation equations (1b) and (9),

$$\begin{aligned} \mathbf {S}_{\mathrm{cons}}\!=\!\! \left[ \begin{array}{l} 0 \\ -\partial _t\pi ^{00} - {{\,\mathrm{div}\,}}(\mathbf {v} P) - (\partial _x\pi ^{0x} + \partial _y\pi ^{0y} + \partial _z\pi ^{0z}) \\ -\partial _t\pi ^{0x} - \partial _xP - (\partial _x\pi ^{xx}+\partial _y\pi ^{xy}+\partial _z\pi ^{xz}) \\ -\partial _t\pi ^{0y} - \partial _yP - (\partial _x\pi ^{yx} +\partial _y\pi ^{yy} + \partial _z\pi ^{yz}) \\ -\partial _t\pi ^{0z} - \partial _zP - (\partial _x\pi ^{zx} + \partial _y\pi ^{zy}+\partial _z\pi ^{zz}) \end{array} \right] ,\nonumber \\ \end{aligned}$$

(A2)

and to Israel–Stewart relaxation equations (6) for viscous fields

$$\begin{aligned} \mathbf {S}_\pi&= \big [Q^{xy}, Q^{xz}, Q^{yz}, Q^{yy}, Q^{zz}\big ]^{\mathrm{T}} \nonumber \\ Q^{\mu \nu }&=\pi ^{\mu \nu }\left( \mathrm{div}\mathbf {v} -\frac{1}{\gamma \tau _\pi }\right) +\frac{\eta }{\gamma \tau _\pi }\,W^{\mu \nu }. \end{aligned}$$

(A3)

In Ref. [43] it was noted that the algorithm could become more stable if the relaxation equations are solved by a simple centered second-order differences scheme for spatial gradients on the left-hand side of Eqs. (6). We have tested such a separation for the full $3+1$D calculations and find out that it leads to uncontrolled solutions. The same phenomenon was observed also for calculation done in the Milne coordinates in Ref. [22], where the authors used also the full SHASTA method for both conservation and relaxation equations. We think that such behaviour is caused by weak steadiness of the Euler method which leads to uncontrolled inaccuracy of a numerical solution of relaxation equations.

Thus we apply the SHASTA method to all ten equations included in Eq. (10). To reach the quadratic precision in time we use Heun’s method [84] which allows storing fewer intermediate points than the mid-point rule.

1.1 1. 3$+$1D implementation of the SHASTA algorithm

For completeness, we provide the complete set of formulas for the 3+1 implementation of the SHASTA algorithm extending expressions provided in Ref [1].

For lattice realization of quantities U(x, y, z, t) we will use notations $U^{[n]}_{ijk}$, where index n stands for temporal steps and i, j, k for spatial lattice cells in x, y, and z directions respectively.

At the first stage of the SHASTA algorithm for the subsequent $(n +1)$th time step, one calculates the so-called transport-diffused solution

$$\begin{aligned} {\widetilde{U}}_{ijk}^{[n+1]}&= {\widetilde{U}}_{ijk}^x + {\widetilde{U}}_{ijk}^y + {\widetilde{U}}_{ijk}^z - 2U_{ijk}^{[n]} + \Delta t\, {S}_{ijk}, \end{aligned}$$

(A4)

where $U_{ijk}^{[n]}$ is the full solution at the previous time step and auxiliary quantities ${\tilde{U}}_{ijk}^{x,y,z}$ are defined as

$$\begin{aligned} {\widetilde{U}}_{ijk}^x&=\frac{1}{2}\left( \big [ Q_{ijk}^{x+} \big ]^2 \big (U^{[n]}_{i+1,jk} - U^{[n]}_{ijk}\big ) \right. \nonumber \\&\left. \quad - \big [Q_{ijk}^{x-}\big ]^2 \big (U^{[n]}_{ijk} - U^{[n]}_{i-1,jk}\big ) \right) +(Q_{ijk}^{x+}+Q_{ijk}^{x-}) \, U^{[n]}_{ijk}, \end{aligned}$$

(A5)

$$\begin{aligned} {\widetilde{U}}_{ijk}^y&=\frac{1}{2}\left( \big [Q_{ijk}^{y+}\big ]^2 \big ( U^{[n]}_{i,j+1,k}- U^{[n]}_{ijk} \big ) \right. \nonumber \\&\left. \quad - \big [Q_{ijk}^{y-}\big ]^2 \big ( U^{[n]}_{ijk}- U^{[n]}_{i,j-1,k} \big ) \right) +(Q_{ijk}^{y+}+Q_{ijk}^{y-}) \, U^{[n]}_{ijk}, \end{aligned}$$

(A6)

$$\begin{aligned} {\widetilde{U}}_{ijk}^z&=\frac{1}{2}\left( \big [Q_{ijk}^{z+}\big ]^2 \big (U^{[n]}_{ij,k+1} - U^{[n]}_{ijk}\big ) \right. \nonumber \\&\left. \quad - \big [Q_{ijk}^{z-}\big ]^2\big (U^{[n]}_{ijk} - U^{[n]}_{ij,k-1}\big )\right) + (Q_{ijk}^{z+} + Q_{ijk}^{z-})\, U^{[n]}_{ijk} \end{aligned}$$

(A7)

with

$$\begin{aligned} Q_{ijk}^{x\pm }&=\frac{1/2\mp \lambda \, (v_x)^{[n]}_{ijk}}{1\pm \lambda \,\left[ (v_x)^{[n]}_{i\pm 1,jk} - (v_x)^{[n]}_{ijk} \right] },\, \nonumber \\ Q_{ijk}^{y\pm }&=\frac{1/2\mp \lambda \, (v_y)^{[n]}_{ijk}}{ 1\pm \lambda \, \left[ (v_y)^{[n]}_{i,j\pm 1,k} - (v_y)^{[n]}_{ijk}\right] },\, \nonumber \\ Q_{ijk}^{z\pm }&=\frac{1/2\mp \lambda \, (v_z)^{[n]}_{ijk}}{1\pm \lambda \,\left[ (v_z)^{[n]}_{ij,k\pm 1} - (v_z)^{[n]}_{ijk} \right] }. \end{aligned}$$

(A8)

The velocity components are taken here at the nth time step. Here, parameter $\lambda =\Delta t/\Delta x=\Delta t/\Delta y=\Delta t/\Delta z$ is the Courant number which is the same for all special directions. In the SHASTA it is restricted to values $\lambda \le 1/2$.

Further, using the transport-diffused solution one calculates an antidiffusion flux that takes into account an anomalous diffusion

$$\begin{aligned} A^{x,y,z}_{ijk}&=\frac{1}{8} A_{\mathrm{ad}}^{x,y,z}\,{\tilde{\Delta }}_{ijk}^{x,y,z},\quad {\tilde{\Delta }}_{ijk}^{x}={\tilde{U}}_{i+1,jk}^x-{\tilde{U}}_{ijk}^x,\quad \nonumber \\ {\tilde{\Delta }}_{ijk}^{y}&={\tilde{U}}_{i,j+1,k}^y-{\tilde{U}}_{ijk}^y,\quad {\tilde{\Delta }}_{ijk}^{z}={\tilde{U}}_{ij,k+1}^z-{\tilde{U}}_{ijk}^z, \end{aligned}$$

(A9)

where $A_{\mathrm{ad}}^{x,y,z}$ are the antidiffusive mask coefficients. For simplicity, one takes them to be equal for all special directions and set $A_{\mathrm{ad}}=1$ as the default value. Next, we calculate the limited antidiffusion fluxes

$$\begin{aligned} \widetilde{A}_{ijk}^x&= \sigma _{ijk}^x \max \Big [0,\min \Big (\sigma _{ijk}^x{\widetilde{\Delta }}_{i+1,jk}^{x},\big |A^x_{ijk}\big |, \sigma _{ijk}^x{\widetilde{\Delta }}_{i-1,jk}^{x}\Big )\Big ],\nonumber \\ \widetilde{A}_{ijk}^y&= \sigma _{ijk}^y \max \Big [0,\min \Big (\sigma _{ijk}^y{\widetilde{\Delta }}_{i,j+1,k}^{y},\big |A^y_{ijk}\big |, \sigma _{ijk}^y{\widetilde{\Delta }}_{i,j-1,k}^{y}\Big )\Big ],\nonumber \\&\qquad \qquad \sigma _{ijk}^{x,y,z}=\mathrm{sgn} A^{x,y,z}_{ijk}. \nonumber \\ \widetilde{A}_{ijk}^z&=\sigma _{ijk}^z \max \Big [0,\min \Big (\sigma _{ijk}^z{\widetilde{\Delta }}_{ij,k+1}^{z},\big |A^z_{ijk}\big |, \sigma _{ijk}^z{\widetilde{\Delta }}_{ij,k-1}^z\Big )\Big ]. \end{aligned}$$

(A10)

The total incoming and outgoing antidiffusive fluxes in the cell are calculated as

$$\begin{aligned} A^{\mathrm{in}}_{ijk}&= \max \big (0,{\widetilde{A}}^x_{i-1,jk}\big ) -\min \big (0,{\widetilde{A}}^x_{ijk}\big ) +\max \big (0,{\widetilde{A}}^y_{i,j-1,k}\big )\nonumber \\&\quad -\min \big (0,{\widetilde{A}}^y_{ijk}\big ) +\max \big (0,{\widetilde{A}}^z_{ij,k-1}\big ) -\min \big (0,{\widetilde{A}}^z_{ijk}\big ), \end{aligned}$$

(A11)

$$\begin{aligned} A^{\mathrm{out}}_{ijk}&= \max \big (0,{\widetilde{A}}^x_{ijk}\big ) -\min \big (0,{\widetilde{A}}^x_{i-1,jk}\big ) +\max \big (0,{\widetilde{A}}^y_{ijk}\big )\nonumber \\&\quad -\min \big (0,{\widetilde{A}}^y_{i,j-1,k}\big ) +\max \big (0,{\widetilde{A}}^z_{ijk}\big ) -\min \big (0,{\widetilde{A}}^z_{ij,k-1}\big ). \end{aligned}$$

(A12)

The maximal and minimal values of the transport-diffused solution $U^{[n+1]}_{ijk}$ after the antidiffusion stage are between

$$\begin{aligned}&{\widetilde{U}}^{\min }_{ijk}= \min \left( {\widetilde{U}}_{ij,k-1}^{[n+1]}, {\widetilde{U}}_{i,j-1,k}^{[n+1]}, {\widetilde{U}}_{i-1,jk}^{[n+1]}, {\widetilde{U}}_{ijk}^{[n+1]},\right. \nonumber \\&\left. \quad {\widetilde{U}}_{ij,k+1}^{[n+1]}, {\widetilde{U}}_{i,j+1,k}^{[n+1]}, {\widetilde{U}}_{i+1,jk}^{[n+1]}\right) , \end{aligned}$$

(A13)

$$\begin{aligned}&{\widetilde{U}}^{\max }_{ijk}= \max \left( {\widetilde{U}}_{ij,k-1}^{[n+1]}, {\widetilde{U}}_{i,j-1,k}^{[n+1]}, {\widetilde{U}}_{i-1,jk}^{[n+1]}, {\widetilde{U}}_{ijk}^{[n+1]},\right. \nonumber \\&\left. \quad {\widetilde{U}}_{ij,k+1}^{[n+1]}, {\widetilde{U}}_{i,j+1,k}^{[n+1]}, {\widetilde{U}}_{i+1,jk}^{[n+1]}\right) . \end{aligned}$$

(A14)

This information is then used to determine the fractions of the incoming and outgoing fluxes,

$$\begin{aligned} F^{\mathrm{in}}_{ijk} = \frac{{\widetilde{U}}_{ijk}^{\max } - {\widetilde{U}}_{ijk}^{[n+1]}}{A^{\mathrm{in}}_{ijk}}, \qquad F^{\mathrm{out}}_{ijk} = \frac{{\widetilde{U}}_{ijk}^{[n+1]} - {\widetilde{U}}_{ijk}^{\min }}{A^{\mathrm{out}}_{ijk}}. \end{aligned}$$

(A15)

The final anti-diffusion fluxes are calculated as

$$\begin{aligned} {\hat{A}}^x_{ijk}&={\widetilde{A}}^x_{ijk} \Big [\min (1, F^{\mathrm{in}}_{i+1,jk},F^{\mathrm{out}}_{ijk})\Theta ({\widetilde{A}}^x_{ijk})\nonumber \\&\quad +\min (1, F^{\mathrm{in}}_{ijk},F^{\mathrm{out}}_{i+1,jk})\Theta (-{\widetilde{A}}^x_{ijk})\Big ], \end{aligned}$$

(A16)

$$\begin{aligned} {\hat{A}}^y_{ijk}&={\widetilde{A}}^y_{ijk} \Big [\min (1, F^{\mathrm{in}}_{i,j+1,k},F^{\mathrm{out}}_{ijk})\Theta ({\widetilde{A}}^y_{ijk})\nonumber \\&\quad +\min (1, F^{\mathrm{in}}_{ijk},F^{\mathrm{out}}_{i,j+1,k})\Theta (-{\widetilde{A}}^y_{ijk})\Big ], \end{aligned}$$

(A17)

$$\begin{aligned} {\hat{A}}^z_{ijk}&={\widetilde{A}}^y_{ijk} \Big [\min (1, F^{\mathrm{in}}_{ij,k+1},F^{\mathrm{out}}_{ijk})\Theta ({\widetilde{A}}^z_{ijk})\nonumber \\&\quad +\min (1, F^{\mathrm{in}}_{ijk},F^{\mathrm{out}}_{ij,k+1})\Theta (-{\widetilde{A}}^z_{ijk})\Big ]. \end{aligned}$$

(A18)

Finally, the full solution for $n+1$ time step is given by

$$\begin{aligned} U_{ijk}^{[n+1]}&={\widetilde{U}}_{ijk}^{[n+1]} +\big ({\hat{A}}^x_{i-1,jk}-{\hat{A}}^x_{ijk}\big )\nonumber \\&\quad +\big ({\hat{A}}^y_{i,j-1,k}-{\hat{A}}^y_{ijk}\big ) +\big ({\hat{A}}^z_{ij,k-1}-{\hat{A}}^z_{ijk}\big ). \end{aligned}$$

(A19)

In Fig. 19 we compare the numerical results of the 3$+$1D SHASTA code with the one-pass method in the time evolution, as given by Eq. (A4) and of the SHASTA code improved by Heun’s method with the exact results of the Bjorken model [42] with the viscosity $\eta =10$ MeV/fm$^2$. The exact solutions for energy density, $\epsilon$, longitudinal velocity, $v_z$ and one component of the viscous tensor, $\pi ^{yy}$ are shown in Fig. 19 by dotted lines for two times elapsed after initialization, $\Delta t=4\,\mathrm{fm}/c$ and $8\,\mathrm{fm}/c$. The solid lines show the results for the one-step 3D SHASTA. We see typical increasing oscillation expanding from the boundaries (with the square-root divergent boundary conditions) inwards the small z regions. This is typical behaviour for algorithms using a single-time step approach. Applying Heun’s method, we obtain much smoother behaviour of the solution shown by dashed lines. Although such scheme works well for model tasks, we observe that for actual 3$+$1D calculations in the cartesian coordinates the algorithm becomes sometimes unstable (in contrast with the 2$+$1D calculation reported in [43]). Also the described entirely-3D approach can lead to problems with anti-diffusion as was noted in [22, 43]. To avoid the instability of the code and other possible troubles, we applied the 3D splitting method for solving the 3D problem.

Fig. 19

Comparison of the exact results for energy density, $\epsilon$, longitudinal velocity, $v_z$ and one component of the viscous tensor, $\pi ^{yy}$ of the Bjorken model [42] with the numerical results of the 3$+$1D SHASTA code with the one-pass method in the time evolution, Eq. (A4), and of the SHASTA code improved by Heun’s method. Calculations are done for the viscosity $\eta =10$ MeV/fm$^2$. The exact solutions are shown two times elapsed after initialization, $\Delta t=4\,\mathrm{fm}/c$ and $8\,\mathrm{fm}/c$

1.2 Implementation of 3D splitting SHASTA for viscous hydrodynamics

We split the 3D task (10) into three sequential 1D propagations

$$\begin{aligned}&\partial _t\mathbf {U}+\partial _x(v_x\mathbf {U})=\mathbf {S}^{\mathrm{1D}}_x \nonumber \\&\partial _t\mathbf {U}+\partial _y(v_y\mathbf {U})=\mathbf {S}^{\mathrm{1D}}_y \nonumber \\&\partial _t\mathbf {U}+\partial _z(v_z\mathbf {U})=\mathbf {S}^{\mathrm{1D}}_z \end{aligned}$$

(A20)

Here, we replace the 3D propagation with three 1D propagations and the source term (A1) is split into three terms corresponding to the propagation along the separate axes. The source terms (A2) and (A3) contain time derivatives and since we replace the 3D propagation by three 1D propagations, we have to include factors 1/3 before the corresponding terms in $\mathbf {S}^{\mathrm{1D}}_{x,y,z}$. As the result, the final expression for the source term responsible for the propagation along axis $k=x,y,z$ is

$$\begin{aligned} \mathbf {S}^{1D}_k&=\Big (0, -\frac{1}{3}\,\partial _t\pi ^{00} - \partial _k (Pv_k)- \partial _k\pi ^{0k}, \nonumber \\&\quad -\frac{1}{3}\,\partial _t\pi ^{0x} - \partial _x P - \partial _k\pi ^{kx}, \nonumber \\&\quad -\frac{1}{3}\,\partial _t\pi ^{0y} - \partial _y P - \partial _k\pi ^{ky} \nonumber \\&\quad -\frac{1}{3}\,\partial _t\pi ^{0z} - \partial _z P - \partial _k\pi ^{kz}, \nonumber \\&\qquad Q^{xy}_k, Q^{xz}_k, Q^{yz}_k, Q^{yy}_k, Q^{zz}_k \Big ), \end{aligned}$$

(A21)

where

$$\begin{aligned} Q^{ij}_k&= \pi ^{ij}\left( \partial _kv_k-\frac{1}{3\gamma \tau _\pi }\right) +\frac{\eta }{\gamma \tau _\pi }\,W^{ij}_k , \\ W^{ij}_k&= \frac{2}{3}\,u^iu^j\,\theta _k-u^i{\mathcal {D}}_k u^j-u^j{\mathcal {D}}_k u^i \\&\quad -\delta _{ik}\partial _k u^j-\delta _{jk}\partial _k u^i,\quad i\ne j,\\ W^{ii}_k&= 2\Big \{\frac{1}{3}\left[ 1+(u^i)^2\right] \,\theta _k-u^i{\mathcal {D}}_k u^i-\delta _{ik}\partial _k u^k \Big \}, \end{aligned}$$

and

$$\begin{aligned} \theta _k&\equiv \frac{1}{3}\,\partial _t\gamma +\partial _ku^k, \end{aligned}$$

(A22)

$$\begin{aligned} {\mathcal {D}}_k u^i&=\frac{1}{3}\,\gamma \partial _t u^i+u^k\partial _k u^i. \end{aligned}$$

(A23)

As we see, after summation over index k

$$\begin{aligned} \sum _k\theta _k=\theta ,\,\, \sum _k{\mathcal {D}}_k u^i={\mathcal {D}}u^i,\,\, \sum _k W^{ij}_k=W^{ij}. \end{aligned}$$

(A24)

we recover $\theta$ and $W^{\mu \nu }$ defined in Eq. (6), and ${\mathcal {D}}u^i= u_\mu \partial ^\mu u^i$.

Every 1D equation in Eq. (A20) is solved using the standard one-dimensional SHASTA method [24,25,26].

If in some fluid cells the relaxation time becomes smaller than the time step, $\gamma \tau _\pi <\Delta t$, then we must obtain the formal solution (11),

$$\begin{aligned} \pi ^{ij}(t_{n+1})&=\big [\pi ^{ij}(t_{n})-\eta W^{ij}\big ]e^{-\Delta t/(\gamma \tau _\pi )}+\eta W^{ij} \end{aligned}$$

(A25)

after completion of the propagation along all three axes. To obtain the correct result in the 3D splitting approach, we start with the full solution at the nth time step, $\pi ^{ij}(t_{n})$, and perform the following sequence of steps in the spatial directions staring,e.g., with the x directions

$$\begin{aligned}{}[\pi ^{ij}(t_{n+1})]_x&= e^{-\Delta t/(\gamma \tau _\pi )} \pi ^{ij}(t_{n})+\big (1-e^{-\Delta t/(\gamma \tau _\pi )}\big )\eta W^{ij}_x, \nonumber \\ [\pi ^{ij}(t_{n+1})]_y&= [\pi ^{ij}(t_{n+1})]_{x}+\big (1-e^{-\Delta t/(\gamma \tau _\pi )}\big )\eta W^{ij}_y, \nonumber \\ [\pi ^{ij}(t_{n+1})]_z&= [\pi ^{ij}(t_{n+1})]_{y}+\big (1-e^{-\Delta t/(\gamma \tau _\pi )}\big )\eta W^{ij}_z, \end{aligned}$$

(A26)

so that after the third step we recover the expected expression.

When one uses 3D splitting method, it is necessary to change the order of 1D propagations to decrease numerical errors. A similar kind of inaccuracy would appear if one uses the same relations among $\pi ^{ij}$ matrix elements, e.g., $\pi ^{xx}=\pi ^{xx}(\pi ^{yy},\pi ^{zz})$ permanently. Therefore, we change the independent spatial diagonal components, $\pi ^{ii}$ and $\pi ^{jj}$, at every time step, see Eq. (B10).

The results of the application of the 3D splitting scheme for the viscous 2nd-order Bjorken expansion are shown in Fig. 19 by dash-dotted lines. The calculations are performed for the same spatial step, $\Delta x=0.2$ fm, as used for the 3D SHASTA and Heun’s-improved 3D SHASTA calculations. We see that the numerical results are very close to the theoretical predictions and fluctuations are weaker than for the Heun’s-method improved algorithm. These fluctuations decrease even further if one takes a shorter step. The results of calculations with $\Delta x=0.1$ fm are shown by dot-dot-dashed lines in Fig. 19. For the most shown quantities, the 3D splitting results are smooth and almost coincide with the exact solutions, only the energy density $\epsilon$ for the later time, $\Delta t=8\,\mathrm{fm}/c$, deviates from the exact solution. This deviation vanishes if we go to a smaller step, e.g., $\Delta x=0.05$ fm, as shown by short dashes. In the actual calculations we have verified on several examples that our results do not change when we reduce the spatial steps from 0.2 fm to 0.1 fm.

Appendix B: Reconstruction of local quantities and exact initialization

The hydrodynamics code evolves the components of the energy-stress tensor and baryon current. The equation of state is formulated in the local system where the energy density and the particle number should be defined. To make a Lorentz transformation from the laboratory frame in the local rest frame one also needs to define a 4-velocity of the fluid element. If we know the components of the ideal stress tensor, $T_{\mathrm{id}}^{\mu \nu }=T^{\mu \nu }-\pi ^{\mu \nu }$ and the baryon current $J^\mu =n\, u^\mu$, other quantities can be recovered as follows:

$$\begin{aligned} n&= J^0/\gamma ,\quad \epsilon =T_{\mathrm{id}}^{00}- M\,v, \nonumber \\ M^2&= T^{0x}_{\mathrm{id}}\,T^{0x}_{\mathrm{id}} + T^{0y}_{\mathrm{id}}\,T^{0y}_{\mathrm{id}} + T^{0z}_{\mathrm{id}}\,T^{0z}_{\mathrm{id}}. \end{aligned}$$

(B1)

The modulus of the fluid velocity can be found as a root of the equation

$$\begin{aligned} v=\frac{M}{T^{00}_{\mathrm{id}}-P\big (T^{00}_{\mathrm{id}}-M\, v,J^0/\gamma \big )} \end{aligned}$$

(B2)

and, therefore, depends on the chosen equation of state $P=P(\epsilon ,n)$. The direction of the fluid velocity is determined as

$$\begin{aligned} v^i=\frac{v}{M}\,T^{0i}_{\mathrm{id}}. \end{aligned}$$

(B3)

Relations (B1), (B2), and (B3) can be used for the ’ideal’ initialization of the hydrodynamic phase when $\pi ^{\mu \nu }=0$ and $T^{0\nu }=T^{0\nu }_{\mathrm{id}}$. For the ’exact’ initialization we have to solve the eigenvalue problem $u_\mu T^{\mu \nu }=\varepsilon u^\nu$, which leads to the quartic algebraic equation

$$\begin{aligned} \varepsilon ^4 + a_1\varepsilon ^3 + a_2\varepsilon ^2 + a_3\varepsilon + a_4=0, \end{aligned}$$

(B4)

where coefficients $a_{1,2,3,4}$ are functions of the energy–momentum tensor invariants,

$$\begin{aligned} a_1&=-T^\nu _\nu =-\text {Tr}\,T^\mu _\nu ,\quad a_2=\frac{1}{2}\left( a_1^2-T^\mu _\nu T_\mu ^\nu \right) , \nonumber \\ a_3&=a_1a_2-\frac{a_1^3+T^\mu _\nu T^\nu _\lambda T^\lambda _\mu }{3}, \nonumber \\ a_4&=-\det T^{\mu \nu }=-\varepsilon ^{\kappa \lambda \mu \nu }T^0_\kappa T^1_\lambda T^2_\mu T^3_\nu . \end{aligned}$$

(B5)

The corresponding velocity is calculated as

$$\begin{aligned} v_z&=\frac{T^{03}(\varepsilon +T^{11})(\varepsilon +T^{22}) -T^{02}T^{23}(\varepsilon +T^{11})-T^{01}T^{13}(\varepsilon +T^{22})+T^{01}T^{12}T^{23}+T^{02}T^{12}T^{13}-T^{03}(T^{12})^2}{(\varepsilon +T^{33})(\varepsilon +T^{22})(\varepsilon +T^{11})-(T^{12})^2(\varepsilon +T^{33}) -(T^{13})^2(\varepsilon +T^{22})-(T^{23})^2(\varepsilon +T^{11})+2T^{12}T^{13}T^{23}}, \nonumber \\ v_y&=\frac{T^{02}(\varepsilon +T^{11})-T^{01}T^{12} +\left[ T^{12}T^{13}-T^{23}(\varepsilon +T^{11})\right] v_z}{(\varepsilon +T^{22})(\varepsilon +T^{11})-(T^{12})^2}, \qquad \nonumber \\ v_x&=\frac{T^{01}-T^{12}v_y-T^{13}v_z}{\varepsilon +T^{11}}. \end{aligned}$$

(B6)

Given the four velocity, $u^\mu =\gamma (1,\mathbf {v})$, we calculate baryon density as

$$\begin{aligned} n&=J^\nu u_\nu , \end{aligned}$$

(B7)

Then knowing the equation of state $P=P(\varepsilon ,n)$, we can define the ideal part of the energy–momentum tensor $T_{\mathrm{id}}^{\mu \nu }$ in Eq. (3). The viscous parts of the full $T^{\mu \nu }$ tensor (2) follow as

$$\begin{aligned} \Pi&=\frac{1}{3}\,\left( \varepsilon -T^\nu _\nu \right) -P, \nonumber \\ \pi ^{\mu \nu }&=T^{\mu \nu }-(\varepsilon +P+\Pi )u^\mu u^\nu +(P+\Pi )g^{\mu \nu } , \end{aligned}$$

(B8)

and the baryon diffusion current as

$$\begin{aligned} V^\mu&=J^\mu -n u^\mu . \end{aligned}$$

(B9)

In the code we use quantities $\pi ^{xy}$, $\pi ^{xz}$, $\pi ^{yz}$, $\pi ^{ii}$, and $\pi ^{jj}$ as independent variables. Other components can be recovered with the help of the following expressions

$$\begin{aligned} \pi ^{0i}&= \pi ^{ik} v_k+\pi ^{ii} v_i+\pi ^{ij} v_j, \nonumber \\ \pi ^{0j}&= \pi ^{jk} v_k+\pi ^{ij} v_i+\pi ^{jj} v_j, \nonumber \\ \pi ^{kk}&= \frac{1}{1-v_k^2} \Big [\pi ^{0i} v_i+\pi ^{0j}v_j+(\pi ^{ik} v_i +\pi ^{jk} v_j )v_k \nonumber \\&\quad -(\pi ^{ii} +\pi ^{jj}) \Big ],\nonumber \\ \pi ^{0k}&= \pi ^{kk}v_k+\pi ^{ik}v_i+\pi ^{jk}v_j, \nonumber \\ \pi ^{00}&=\pi ^{ii}+\pi ^{jj}+\pi ^{kk}. \end{aligned}$$

(B10)

We emphasize that these expressions do not develop anomalously large values for the case of small fluid velocities.