Entropy-limited hydrodynamics: a novel approach to relativistic hydrodynamics

We first show the accuracy of the scheme in the case of a smooth solution and measure rigorously its convergence order so as to show that the entropy-driven limiter does not spoil the convergence properties of the high-order method upon which it is built. This test has been discussed in Radice and Rezzolla (2012) (which adapted it from Zhang and MacFadyen 2006). In short, we consider a one-dimensional, large-amplitude, smooth, nonlinear wave with initial rest-mass density profile given by where \(L=0.3\). The initial data employs a polytropic EOS, \(p=K\rho^{\tilde{\gamma}}\), with \(K=100\) and \(\tilde{\gamma}=5/3\), and we then evolve it with the ideal-fluid EOS (4) with \(\Gamma =5/3\). Since in this test discontinuities are absent (so that \(\tilde{\gamma }=\Gamma\)) and there are no stability issues, we use as time integrator the standard fourth-order RK4 method with a timestep of ∼0.13 times the grid spacing.

The analytic solution of the test (shown in Figure 1 with a black solid line) is represented by a wave profile propagating towards the right and ‘steepening’ in the direction of its motion. At time \(t_{c}\simeq1.6\), the wave develops a shock, leading to a sharp discontinuity. Up to the formation of the caustic, the analytic solution can be computed using the method of characteristics (Anile 1990) on a Lagrangian grid. We obtain an accurate enough approximation by computing it on a very fine grid of 10⁵ gridpoints and interpolating the solution using cubic splines on the Eulerian grid. This solution, which we refer to as the ‘exact’ solution, is then used as the reference against which the numerical solutions are compared.

We perform this test with both the EL5 and EL7 schemes of the ELH method to validate that high-order schemes can be employed with great ease in our approach by simply swapping a lower-order stencil for a higher-order one; this operation is far more demanding in standard finite-volume HRSC schemes.

Figure 2 shows the \(L_{1}\)-norm of the error with respect to the analytic solution at time \(t=0.8\) for the various schemes and at various resolutions. The latter are parametrized by the number of gridpoints used on the x axis, and we have considered seven different resolutions, each twice as fine as the preceding one, going from 100 gridpoints up to \(6\text{,}400\). The different lines in Figure 2 show that at the lowest resolutions all schemes show very similar errors, MP5 being the most accurate by a small margin. As the resolution is increased, however,the gap in accuracy between EL5 and MP5 decreases and disappears at very high resolutions. The error curve of EL7, being a higher-order scheme, decreases much more rapidly with resolution, so that its error at the highest resolution of \(6\text{,}400\) gridpoints is two orders of magnitude lower than for the fifth-order schemes.

We also compute the convergence order of the various schemes using the data at resolutions of \(1\text{,}600\) and \(3\text{,}200\) gridpoints and after comparing it with the ‘exact’ solution. The result is shown in Figure 3 as a function of time to the development of the shock. The computed order should be equal to the nominal order of each scheme as long as the solution is smooth, gradually degrading to first order as the caustic is approached. Every scheme matches this description, in particular EL5, whose convergence order is almost exactly five. EL7 similarly appears to saturate just below its nominal convergence order of seven. Deviations from the nominal convergence order of each scheme are due to contaminations from other error sources, which become increasingly significant at high resolution; these are: the truncation error due to the time-integrator, the accuracy of the inversion from conservative to primitive variables, or the low-order approximation for the evolution of the entropy (cf., Eq. (22)). What is relevant here is that the EL method does not interfere with the convergence properties of the underlying stencil and can exploit their accuracy.

We choose as a first shock-tube test the special-relativistic version of the classical Sod test (Sod 1978). In this case, the adiabatic index for both the polytropic initial data EOS and the ideal-fluid evolution EOS is \(\Gamma=1.4\) and the right (R) and left (L) initial states are The analytic solution consists in a left-going rarefaction wave and a right-going shock wave separated by a right-going contact discontinuity. We perform the test with a variety of spatial resolutions ranging from \(\Delta x=0.01\) to \(\Delta x=3.125 \times10^{-4}\), and a timestep \(\Delta t=0.1~\Delta x\).

Figure 4 shows the test results at time \(t=0.6\) for both the EL5 and MP5 schemes at resolution \(\Delta x=1.25 \times 10^{-3}\). Both schemes capture the main features of the solution, with the shocks being captured within ∼3 gridpoints, as are the constant states in the pressure and velocity. However, the EL5 scheme displays some oscillations downstream of the shock as well as over- and undershoots around the location of the discontinuities and in the transition between the rarefaction wave and the surrounding flat regions, while the MP5 scheme is able to resolve the solution avoiding such artefacts. This is not surprising since MP5 is a monotonicity preserving scheme (the number of local maxima and minima cannot increase by effect of this method, therefore over- and undershoots cannot occur by construction) while EL5 is not. We should remark, however, that this property of MP5 is valid only for scalar equations in one spatial dimension, and it basically accounts for the differences in the behaviour of the two schemes. It should also be stressed that the EL5 scheme is indeed stable and that the oscillations that are present in the solution converge away with resolution (see Figure 5).

The behaviour of the viscosity is displayed in Figure 6. It presents four well distinct peaks, each corresponding to the four nonlinear waves generated by the Riemann problem and corresponding to the edges of the rarefaction wave, where the solution is continuous but non-smooth, of the contact discontinuity and of the shock. The viscosity is higher in correspondence with the latter, and decreases by several orders of magnitude for the other three. It can also be seen clearly how the peaks in the viscosity sharpen as the resolution is increased, mirroring the decreasing size of the aforementioned features (see Figure 5), and seemingly tending towards a delta function at infinite resolution, as expected.

The second shock-tube test we select is a more extreme ‘blast-wave’ test (Martí and Müller 2003). In this case, the adiabatic index used is \(\Gamma=5/3\) and the right and left initial states are The exact solution consists in a right-going shock wave, followed by a contact discontinuity and a left-going rarefaction wave. We employ the same resolutions and timestep choices as for the Sod test.

Figure 7 is similar to Figure 4, but relative to the blast-wave test at time \(t=0.4\). We should remark that this is a very extreme test (the pressure has an initial jump of five orders of magnitude) in which the contact discontinuity and shock wave move at essentially the same speed, yielding a very narrow constant rest-mass density state between the two. The oscillations in the EL5 scheme data are in this case more severe than in the Sod shock-tube test, especially around the shock location. As a result, the solution with the EL5 scheme tends to a general decrease of the pressure between the rarefaction wave and the shock wave, whose relative value is however of \({\lesssim}7\%\) at most; the MP5 scheme performs better and has a relative error in pressure that is \({\sim}1\%\). In both cases, the agreement with the exact solution improves with resolution.

Finally, we perform a three-dimensional shock-tube problem, involving non-grid-aligned shocks i.e., the relativistic-explosion test. The initial data in this case is given by where r is the distance from the origin. The computational domain is a cube of side 1 centered on the origin, and we use a grid spacing of \(\Delta x=0.01\) and a timestep \(\Delta t=0.1~\Delta x\). The adiabatic index for this test is again \(\Gamma=1.4\). The feature of the solution are similar to those of the Sod test i.e., an ingoing rarefaction wave and an outgoing shock, separated by an outgoing contact discontinuity. Note however that because of the spherical symmetry of the test (compared to the planar symmetry in the Sod case), the regions at the two sides of the contact discontinuity are no longer constant states in rest-mass density, velocity and pressure, but display a smooth radial dependence.

Figure 8 shows the rest-mass density for this test at time \(t=0.25\), on the \((x,y)\) plane as well as on the x axis. Both EL5 and MP5 perform very similarly, with differences being barely noticeable in the two-dimensional plot. The curves on the x axis reveal that while both schemes capture the features of the solution, as in the Sod test, the EL5 scheme is slightly more oscillatory.

Overall, these shock-tube tests demonstrate how the entropy-driven hybridisation of the high-order stencil is sufficient to stabilise the scheme even for discontinuous initial data and it is remarkable that such a simple scheme can achieve good accuracy.

The first test we perform is the evolution of a stable nonrotating (or TOV, from Tolmann-Oppenheimer-Volkoff) neutron star in a fixed spacetime (i.e., adopting the Cowling approximation) with the goal of assessing the properties of the EL5 scheme over long timescales. Despite its conceptual simplicity (a TOV is just a static solution of the Einstein-Euler equations) the test can be rather challenging. This is because in this test the location of the stellar surface, which is the hardest feature to simulate due to the steep gradient in the hydrodynamics variables, is essentially stationary; as a result, errors can accumulate and grow, affecting the accuracy of the simulation. This behaviour is to be contrasted with the typical situation encountered when evolving inspiralling binary neutron stars, where the stellar surfaces move very supersonically with respect to the floor and most of the errors at the surface are absorbed into the shocks.

For this test we build and evolve a TOV model with central rest-mass density \(1.28 \times10^{-3}~M_{\odot}^{-2}\), yielding a (baryon) rest mass of \(1.5~M_{\odot}\) and a radius of \({\sim}10~M_{\odot}\). We perform the test on a single refinement level with outer boundaries placed at \(16~M_{\odot}\) and a resolution of \(\Delta^{i}=0.2~M_{\odot}\simeq0.3~\mbox{km}\). The timestep is set to 0.15 times the grid spacing, and the time integrator is RK3.

Figure 9 reports the distribution of the viscosity on the equatorial plane, which clearly shows a local annular peak around the location of the stellar surface, where the hydrodynamical variables experience the most violent variations, leading to large values of the viscosity. In the external low-density fluid, the viscosity is set to a small constant value almost everywhere as detailed in Section 3.2. The inner part of the neutron star is expected to be isentropic, as it consists of a shock-free perfect fluid. Indeed, in the stellar interior the viscosity is nonzero but also 10² to 10³ times smaller than at the surface and does not significantly affect the evolution. Quite generally, these features of the viscosity profile are typical in all the tests we considered, whenever a sharp matter/vacuum interface is present.

The general behaviour of the EL5 scheme, when compared to the MP5 scheme, is well illustrated by Figure 10, showing the rest-mass density distribution on the equatorial plane for the two schemes (the left part of the panel, i.e., for \(x<0\), refers to the MP5 scheme, while the right part, i.e., for \(x>0\), to the EL5 scheme³). As it can be seen from the figure, both schemes accurately capture the solution in the stellar interior, but significant differences arise at the surface and in the exterior. The MP5 scheme shows a rather diffusive behaviour, with a smooth transition to the external ‘vacuum’ (i.e., to a region close to the rest-mass density floor) and extended low-density tails. The EL5 scheme, on the other hand, produces a sharper edge. Oscillations in the solution can be seen just outside of the star, resulting in shell-like structures around the surface, which are particularly noticeable in the coordinate axes directions. The stellar exterior is much closer to the vacuum with the EL5 scheme and, in contrast to MP5, it also displays small-scale dynamics at very low densities.

The properties of the oscillations present in the solution computed with the EL5 scheme are made clearer in Figure 11, which shows the rest-mass density profiles along different radial cuts. Along the x direction, the oscillations in the EL5 data have large amplitude and a similar behaviour is observed along the y and z axes. On the other hand, on the three-dimensional diagonal (i.e., along the \(x=y=z\) line), the EL5 scheme manages to capture the sharp transition between the stellar interior and the outside vacuum almost perfectly, without significant oscillations or other artefacts. By contrast, the use of the MP5 scheme leads to smooth, rest-density profiles that are only slowly decaying in all directions.⁴

The direction-dependent behaviour shown in Figure 11 for the EL5 scheme is due to the well-known anisotropy of the phase error common to finite-differencing schemes (Vichnevetsky and Bowles 1982; Lele 1992). The MP5 scheme is able to mask this behaviour, but at the price of sacrificing the ability to sharply define stellar surface. We expect that the performance of the EL5 scheme could be improved through the use of multidimensional stencils (i.e., employing a multidimensional interpolation in the reconstruction step), as opposed to the current approach in which the stencil is simply oriented in the direction of the flux to be reconstructed.

The quantitative differences between the two schemes are better captured in Figure 12, where the evolution of the total rest mass and of the central rest-mass density are shown. We recall that the total rest mass (or baryon mass), is defined as where the integral is performed over the whole computational domain. From the continuity equation (35) follows that it should be conserved in absence of a net total flow of matter in or out of the domain. The numerical schemes we employ are conservative (see e.g., Leveque 1992), and therefore preserve the value of the rest mass to the one determined by the initial data. Nonetheless, violations of this conservation can take place in at least three different ways. First, winds originating at the stellar surface (physically, as e.g., in binary neutron star merger, or spuriously as in a stationary case such as the present one) can yield a net loss of mass when they reach the outer boundary and leave the computational domain. Second, matter can be spuriously created or destroyed, in a way that is hard to control, because of floating-point or interpolation errors at the boundaries of refinement levels (this is not the case for this particular test clearly, since we employ a single grid, but it is relevant for the following ones). Finally, when a value of the density is floored mass is spuriously created or destroyed. It is therefore important to characterize the interplay between the numerical scheme and these grid related effects.

The left panel of Figure 12 shows deviations, in absolute value, of the rest mass from the initial value for the two schemes. The EL5 scheme is evidently much better at conserving mass in this test than MP5, leading to a cumulative deviation of \({\sim}10^{-7}~M_{\odot}\) which is almost three orders of magnitude smaller than the MP5 value.

The central rest-mass density also undergoes an evolution (right panel of Figure 12), with oscillations triggered by the treatment of the stellar surface. Both schemes perform at a similar level of accuracy, with relative variations from the initial value no greater than about 0.3% (even though spurious peaks are present in both data series at various times). The short term behaviour of the two schemes is noticeably different, and the frequency content in the two data series appears different, with the MP5 scheme seeming to show more pronounced high-frequency modes. However, at later times both schemes appear to relax and oscillations decrease significantly in amplitude.

We then proceed to relax the Cowling approximation and test the entropy-limited method coupled with a dynamically evolved spacetime. As first step, we evolve the same initial data for a isolated stable star as in the previous section (i.e., with central density \(1.28 \times10^{-3}~M_{\odot}^{-2}\), baryon mass of \(1.5~M_{\odot}\) and radius \({\sim}10~M_{\odot}\)). We perform the test on a grid consisting of three refinement levels centered on the star with sides lengths 16, 32 and \(60~M_{\odot}\) from finest to coarsest, and with a constant refinement factor of 2. The spatial resolution of the innermost and finest level is set to \(\Delta^{i}=0.2~M_{\odot}\simeq0.3~\mbox{km}\), and the timestep to 0.15 times the grid spacing. This factor is largest possible to guarantee the positivity of the rest-mass density (see discussion in Section 3.1 and Radice et al. 2014a for details). The atmosphere value of the density is set to \(10^{-16}~M_{\odot}^{-2}\), that is, almost 13 orders of magnitude smaller than the maximum value. As a time integrator we select the third-order SSP Runge-Kutta RK3. Unless stated differently, we employ the same grid setup for each one of the following single star tests.

Figure 13 shows the distribution of rest-mass density on the equatorial plane for this test, again with the MP5 and EL5 schemes shown on the left and right parts of the panel, respectively. It can be appreciated how the MP5 scheme produces rest-mass tails which are even more dense and extended than in the Cowling case, making the near vacuum solution obtained by the EL5 scheme all the more striking.

Another difference from the Cowling test can be seen in the conservation of the rest mass (left panel of Figure 14). In this case too, the EL5 scheme is able to conserve the initial value to an accuracy roughly two orders of magnitude better than the MP5 scheme. Furthermore, it is interesting to notice how the behaviour of the EL5 scheme is much more smooth and predictable; MP5 by contrast displays both spurious losses and gains of mass, which lead to the zero crossings clearly visible in the figure. This is due to interpolation errors arising during the restriction and prolongation operations between different refinement levels. These errors are more severe with MP5 due to the presence of long tails of low density matter in the stellar exterior, as we checked by varying the extent of the refinement levels. In contrast, the EL5 scheme is less affected since the exterior of the star (especially away from the coordinate axes) is nearly vacuum.

The evolution of the central rest-mass density, as shown in the right panel of Figure 14, is similar to the one shown in the previous section for the Cowling approximation, with both schemes varying no more than 0.2% from the initial value, but with MP5 displaying oscillations at much higher frequency.

To further investigate this point, we compute the power spectral density (PSD) of the density evolution, in order to quantitatively gauge the differences between the two schemes. The PSD is computed over the first \(5\text{,}000~M_{\odot}\) of data and with the use of a Hann window function. Before computing the PSD, any linearly growing component of the signal is removed via a least-squares fit.

The PSDs for both schemes are shown in Figure 15 along with the oscillation frequencies of this stellar model as computed perturbatively following the methods discussed in Yoshida and Eriguchi (2001), Takami et al. (2011), and shown as vertical dashed lines. For both schemes, the PSD is dominated by a low-frequency component due the well-known secular changes in the central rest-mass density (Font et al. 2002) and that disappear with resolution. However, peaks are clearly visible above the noise. The lowest-frequency peaks correspond to the fundamental oscillation mode of the star and its first overtone, while the following ones are higher overtones and are progressively more offset from the corresponding perturbative eigenfrequencies. The peaks in the EL5 data appear to be more clearly identifiable and less broad than in the MP5 case. Above \({\sim}8\text{,}000~\mbox{Hz}\), and not shown in Figure 15, the MP5 scheme shows significant high-frequency components, clearly visible in the first part of the corresponding curve in Figure 14, but with a rather disordered spectrum. These same frequencies are instead greatly suppressed in the EL5 scheme. Overall, also this test highlights that the EL5 scheme captures quite well the physical behaviour of the system as expected from perturbative methods and is free from some of the artefacts which appear instead in the evolution with the MP5 scheme.

The next test we perform is a slight modification of previous setup, i.e., we evolve the same isolated neutron star model, but applying a small velocity perturbation to the initial solution. The perturbation consists of a radially outgoing velocity growing linearly in radius to a maximum value of 0.005.

We employ this scenario, more realistic than the simple smooth-wave test of Section 4.1.1, to measure the convergence order of the EL5 and MP5 methods. We performed three sets of simulations at resolutions 0.24, 0.12 and \(0.06~M_{\odot}\) on the finest level, extracting the evolution of the rest-mass density over time from each one. The initial perturbation is added so that the density evolution is not dominated by the truncation error, but displays a cleaner behaviour. Otherwise, as the resolution is increased, the density evolution would show additional high-frequency modes, which would make the dependence on resolution discontinuous, making it difficult to compute the instantaneous convergence order.

Using the values of the \(L_{1}\)-norm of the rest-mass density over the domain at the three resolutions we also computed the instantaneous convergence order \(M_{\odot}\) as shown in Figure 16. Because this is the instantaneous convergence order and because the underlying system is oscillating, the curves are somewhat noisy (especially for MP5); however, when taking the running average, both schemes generally show a convergence order just below three, consistent with the results in Radice et al. (2014a, 2014b). It is also however apparent how EL5 maintains a fairly constant order of convergence through time, while the behaviour of MP5 is more irregular, especially at later times.

While the hydrodynamics schemes are both formally fifth-order accurate, other parts of the algorithm operate at different degrees of accuracy. In particular, the time integrator is third-order accurate, which most likely accounts for the convergence order being closer to three than to five. The result is also consistent with the ones found for the MP5 scheme in Radice et al. (2014a, 2014b), Radice et al. (2015). Overall, this test highlights how both the ELH and MP5 schemes perform fairly consistently over time, with no major loss of accuracy.

Another important test in our series is the migration of a TOV star moving from a solution on the unstable branch of equilibrium solutions to a stable one. We recall that for any given EOS, increasingly massive but stable TOV models can be constructed by considering increasingly large values of the central rest-mass density. This can continue until a maximum mass is reached, at which point, an increase of the central rest-mass density corresponds to a decrease of the mass of the star. Models on this second branch of the mass/central-rest-mass density curve are unstable, and if a perturbation is present will evolve to either a stable configuration or collapse to a black hole. This is precisely the physical scenario that the migration test simulates: we construct a model on the unstable branch of the mass/density curve and force its ‘migration’ to a stable configuration by applying a suitable velocity perturbation.

This is a common test for numerical relativity codes (see, e.g., Font et al. 2002; Baiotti et al. 2003, 2005; Cordero-Carrión et al. 2009; Thierfelder et al. 2011), and has been studied in detail in Radice et al. (2010). In particular, we build a nonrotating stellar model on the unstable branch of the equilibrium solutions and with central rest-mass density of \(7 \times10^{-3}~M_{\odot}^{-2}\) (yielding a total rest mass of \(1.6~M_{\odot}\) and a radius of \(6~M_{\odot}\)). The migration is then triggered by injecting a radially outgoing velocity perturbation where the velocity grows linearly in radius, reaching a maximum value of 0.01. The star then undergoes a series of violent expansions and contractions as it migrates to the stable branch and then settles on the new equilibrium. During each contraction and expansion strong shocks are formed, and the shocked matter is ejected at large velocities.

In Figure 17 we show the rest-mass density distribution on the equatorial plane for both schemes and during one of the contractions of the star, just before the central rest-mass density reaches a maximum (cf., Figure 18). The snapshot clearly shows that both the EL5 and MP5 schemes produce almost identical results for this test. This is not surprising and mainly due to the matter outflow driven by the stellar oscillations, which rapidly fills the domain and removes the sharp feature of the stellar surface, which is the most problematic structure to resolve and the main difference in the two schemes.

The evolutions of the maximum rest-mass density are shown in Figure 18, where they are reported as normalized to the initial value. The agreement between the two schemes is extremely good during the entire evolution and the main difference between the two solutions is the presence of some high-frequency modes near the maxima of the density in the EL5 data. Such oscillations are the result of inward-propagating shock waves generated in the outer layers of the star during the contraction phase. Figure 19 shows a magnification of the behaviour of the maximum rest-mass density at the peak of the first contraction, comparing not only the two schemes but also the evolutions with two different resolutions. At high resolution, both the MP5 and the EL5 scheme show small-scale and high-frequency oscillations that are less pronounced in the low-resolution data. Interestingly, these oscillations are essentially smoothed out in the low-resolution run of the MP5 scheme, while they are very visible in the low-resolution EL5 run. This seems to indicate that the two schemes tend, with increasing resolution, towards a solution where the small-scale oscillations are present and therefore physically correct and not a numerical artefact. Finally, as the evolution progresses, the contraction/expansion phases become less and less violent as part of the kinetic energy is converted into internal energy, thereby leading to milder and milder shocks, and the high-frequency oscillations in the central rest-mass density all but disappear.

As the last test case for a stable (or metastable) isolated relativistic stars we consider the evolution of a rapidly and uniformly rotating star. More precisely, we set up axisymmetric initial data relative to a uniformly rotating neutron star governed by a polytropic EOS with \(K=100~M_{\odot}^{-2}\) and \(\Gamma=2\), having a central rest-mass density of \(1.28 \times10^{-3}~M_{\odot}^{-2}\) and a polar to equatorial axis ratio of 0.8 using the RNS code (Stergioulas and Friedman 1995). This results in a star with total rest mass \(1.6~M_{\odot}\), radius \(10~M_{\odot}\), rotation frequency \(f=673.2~\mbox{Hz}\) (about 60% of the mass shedding frequency) and dimensionless angular momentum \(J/M^{2}=0.46\). Also in this case the spacetime is evolved in time despite the solution being stationary.

In Figure 20 we report again the rest-mass density distribution on the equatorial plane for both schemes and at time \(t=4\text{,}300~M_{\odot}\), that is, after about 14 rotation periods. The figure clearly illustrates that that both schemes evolve the rotating star with no noticeable problems and that, as already seen in the case of nonrotating stars, the part of the domain exterior to the stellar surface rapidly fills with matter. Also in this case, the behaviour of the two methods in the low-density regions is rather different, with the MP5 scheme yielding to a volume which is filled by uniform but comparatively higher-density material, while the EL5 scheme produces a stellar exterior which has lower-density matter but with small-scale condensations (cf., Figures 10 and 13 for the equivalent behaviour in the absence of rotation).

However, in contrast with the behaviour seen in the case of nonrotating stars, the dynamics of the low-density material in the stellar exterior results in a degradation of the conservation of mass for the EL5 scheme, as shown in the left panel of Figure 21 (cf., the left panels of Figures 12 and 14 for the equivalent behaviour in the absence of rotation). The deviation of the total rest-mass density from its initial value is more than one order of magnitude larger for the EL5 scheme than for MP5 and reaches values of \({\sim}10^{-5}~M_{\odot}\). This is the result of failures in the conversion from the conserved variables to the primitive ones, triggered by oscillations in the solution: the solution can be evolved to an unphysical state, and in this case the rest-mass density could reach values below the atmosphere floor value; if so, the conversion routine resets the affected cells to the atmosphere value, thereby spuriously creating mass. We speculate that most of these failures result from the large tangential velocity that is acquired by the shell-like distribution of matter that builds up in the case of the EL5 scheme and that is present already in the nonrotating case. While rather innocuous in the absence of rotation, this shell of matter can fling material to large distances (but within the computational domain) and lead to a much more chaotic dynamics of the fluid in the low-density regions (see the discussion in Section 3.2.3 of Radice et al. 2014a).

To assess the impact of the fluid dynamics in the stellar exterior we report the evolution of the central rest-mass density for the two schemes in the right panel of Figure 21. We find that the low-density fluctuations appearing in the stellar exterior with the EL5 scheme do not impact the solution in the stellar interior, with the low-frequency central density oscillations essentially being in phase for the two schemes. Also quite apparent is that the EL5 scheme yields rather constant-amplitude oscillations and this should contrasted with the MP5 scheme, where the oscillations are comparatively larger in the first \({\sim}2\text{,}000 M_{\odot}\) of the evolution. In both cases, however, the oscillations are extremely small and below 0.1%.

We further test the ELH scheme in another truly dynamical test: the motion across the numerical grid of two neutron stars in a grazing collision. This is a setup that is very similar to that of a binary-neutron star system in quasi-circular orbit, the most obvious difference being the initial momenta of the two stars do not result in a quasi-circular orbit and that the initial fluid velocity can be taken to be arbitrary. In practice, the initial data is set up by generating two identical TOV models (the same as considered in Sections 4.2.1 and 4.2.2), superimposing the two data sets on the computational grid and imparting suitable initial momenta resulting in a small, but nonzero, impact parameter. Clearly, such initial data is valid only as a first approximation since the stars are not in the hydrostatic equilibrium corresponding to the binary system and the intial metric and extrinsic curvature do not reflect a solution of the Einstein constraints equations.

These violations lead to initial oscillations in the evolution (see Kastaun et al. 2013; Tsatsin and Marronetti 2013 for a more detailed discussion of a more sophisticated setup in which the stars are also subject to a spin up) which can however be reduced significantly by setting the initial distance of the two stars to a rather large value. More importantly, however, these oscillations do not interfere with the main goal of this test, namely, that of validating the ability of the ELH scheme to preserve sharply the features of the stellar surface also when the star moves across the numerical grid.

More in detail, the star centers are set at positions \((x_{1},y_{1},z_{1})=(50,-50,0)\) and \((x_{2},y_{2},z_{2})=(-50,50,0)\) in units of \(M_{\odot}\), i.e., symmetric with respect to the grid center on the \((x,y)\) plane and at a distance of \({\sim}141~M_{\odot}\). The initial 3-velocities are \((v^{x}_{1},v^{y}_{1},v^{z}_{1})=(0,-0.1,0)\) and \((v^{x}_{2},v^{y}_{2},v^{z}_{2})=(0,0.1,0)\) respectively. We evolve the system on a cubic grid of radius \(512~M_{\odot}\), but employ reflection symmetry boundary conditions across the \((x,y)\) plane and 180 degrees rotation symmetry boundary conditions across the \((y,z)\) plane to reduce the computational cost. The grid structure consists of two identical box-in-box refinement levels hierarchies with refinement factor 2, each centered on a star and consisting of 5 cubic levels with radii \(12,25,50,100,200~M_{\odot}\), plus the coarse base level with radius \(512~M_{\odot}\), so that the grid spacing in the innermost refinement level is \(\Delta^{i}=0.2~M_{\odot}\simeq0.3~\mbox{km}\). The refinement levels moved to track the positions of the stars during the evolution (see also Radice et al. 2016 for further details on the initial data and grid structure). We set again \(\Delta t=0.15~\Delta x\).

The evolution consists in the two stars initially traversing the grid in the x direction and approaching each other, then bending their trajectories as in a gravitational scattering process; we do not follow the dynamics of the process after the first fly-by. Figure 22 shows the rest-mass density distribution on the \((x,y)\) plane for this test. The snapshots of one of the two stars are taken at time \(t=768~M_{\odot}\), when the two stars are past the point of closest approach and are escaping. One can appreciate the deformation due to the boost, the acquired spin angular momentum and the tidal gravitational interaction. From the hydrodynamics point of view the behaviour of the MP5 and EL5 schemes is consistent with the previous tests, in particular the TOV in a dynamical spacetime: EL5 shows a sharper star surface with respect to MP5, as well as an ‘emptier’ surrounding region, while the bulk of the star itself is very well resolved by both schemes.

The conservation of the total rest mass (left panel of Figure 23) is instead similar to the rotating-star case, thus showing a better performance by the MP5 scheme. Note however that the differences between the schemes are far smaller in this case, less than one order of magnitude. Furthermore, in contrast with the preceding tests, the grid structure in this case is much more complicated as well as dynamically updated to track the stars. Interpolation errors at the refinement level boundaries play therefore a greater role in the conservation of rest mass.

The evolution of the rest-mass density at the star centers, as shown in the right panel of Figure 23, is very similar for both schemes. There is an initial sudden increase in the density of about 4% with respect to the initial value, due to the evolution scheme bringing the star in hydrostatic equilibrium from the initial state. The density then oscillates around this new value, due to perturbations that in this case are not only induced by the violation of the constraint equations but also by the gravitational interaction. Overall, both schemes reproduce well all of these effects and show a very good agreement.

As a final test we evolve the violently dynamical collapse of a star to a black hole. This is also a common numerical-relativity benchmark (see, e.g., Font et al. 2002; Baiotti et al. 2005; Baiotti and Rezzolla 2006; Thierfelder et al. 2010), which allows us to validate ELH in the presence of a physical singularity and of an apparent horizon. More specifically, we consider a nonrotating star with central rest-mass density \(8 \times10^{-3}~M_{\odot}^{-2}\), corresponding to a baryon mass of \(1.5~M_{\odot}\) and radius \(6~M_{\odot}\), and initiate the collapse with a velocity perturbation analogous to the one used in the migration test, but with the opposite sign, i.e., radially ingoing.

We define the time of black-hole formation as the instant at which an apparent horizon is first detected on the numerical domain, which, given the chosen setup, happens at \(t\simeq48~M_{\odot}\). Since we use singularity avoiding slicing conditions, we do not need to excise the interior spacetime of the black hole (Baiotti and Rezzolla 2006; Baiotti et al. 2007; Thierfelder et al. 2010). At the same time, we set the hydrodynamical variables to their atmosphere values inside a surface with the same shape as the apparent horizon, but radius \(r=0.9~r_{\mathrm{AH}}\) in every angular direction, \(r_{\mathrm{AH}}\) being the radius of the apparent horizon. This ‘hydrodynamic excision’ is not strictly necessary as our code can handle the collapse without it, regardless of the scheme we employ. However, we have observed that its use improves the accuracy of the subsequent evolution, most notably, it improves the behaviour of the rest-mass density and we therefore choose to employ it.

Figure 24 shows a snapshot of the rest-mass density on the equatorial plane just after the collapse. The central area of uniform low density is where the hydrodynamical variables have been set to atmosphere inside the horizon, and in this plot appears of identical size and shape for the two codes. The areas outside the horizon are instead filled with matter which has been spuriously ejected from the outer layers of the star star during the collapse. The rest-mass density is evidently higher in the case of the EL5 scheme, due to a slightly higher proportion of matter ejected during the collapse, which is in turn triggered by larger oscillations around the stellar surface for the EL5 scheme. However, in both cases the total rest mass outside the horizon is tiny, \({\sim}10^{-6}~M_{\odot}\) for the EL5 scheme and \({\sim} 10^{-9}~M_{\odot}\) for MP5, and thus dynamically essentially irrelevant on the properties of the solution. Furthermore, most of this matter is gravitationally bound and hence accretes back onto the newly formed black hole, forming streams of infalling matter. This is particularly evident along the coordinate directions, where the numerical viscosity of the high-order finite-differences stencil is smaller, independently of the scheme employed.

The very close agreement between the two solutions is summarised in Figure 25, where the evolution of the central rest-mass density, minimum lapse and \(L_{2}\)-norm of the Hamiltonian constraint is plotted. The discontinuities in the curves at about \(50~M_{\odot}\) correspond to the time of collapse and arise because we exclude points inside the horizon in the calculation of extrema and norms. In each panel before the formation of the apparent horizon, the curves corresponding to the EL5 and MP5 schemes are essentially on top of each other (the largest differences being of the order of 0.3%, 0.6%, and 4.7% for each plot, respectively), showing the very good agreement in the evolution between the two schemes. Note also that after the apparent horizon formation and due to our approach of ‘hydrodynamic excision’, the upper panel of Figure 25 shows the maximum of the density in the exterior of the horizon rather than the central density. The disagreement in the EL5 and MP5 curves relates therefore to the tiny amount of residual matter outside of the black hole, and as such it has no dynamical impact on the black-hole solution.

A final confirmation of the equivalence between the two numerical solutions comes from the comparison of the two black-hole masses computed using the dynamical horizon formalism (Ashtekar and Krishnan 2003) and shown in Figure 26. As can be seen from the figure, we find a very close agreement between the two schemes.