Entropy-limited hydrodynamics: a novel approach to relativistic hydrodynamics
- Federico Guercilena^{1}Email author,
- David Radice^{2, 3} and
- Luciano Rezzolla^{1, 4}
https://doi.org/10.1186/s40668-017-0022-0
© The Author(s) 2017
Received: 19 December 2016
Accepted: 20 June 2017
Published: 4 July 2017
Abstract
We present entropy-limited hydrodynamics (ELH): a new approach for the computation of numerical fluxes arising in the discretization of hyperbolic equations in conservation form. ELH is based on the hybridisation of an unfiltered high-order scheme with the first-order Lax-Friedrichs method. The activation of the low-order part of the scheme is driven by a measure of the locally generated entropy inspired by the artificial-viscosity method proposed by Guermond et al. (J. Comput. Phys. 230(11):4248-4267, 2011, doi:10.1016/j.jcp.2010.11.043). Here, we present ELH in the context of high-order finite-differencing methods and of the equations of general-relativistic hydrodynamics. We study the performance of ELH in a series of classical astrophysical tests in general relativity involving isolated, rotating and nonrotating neutron stars, and including a case of gravitational collapse to black hole. We present a detailed comparison of ELH with the fifth-order monotonicity preserving method MP5 (Suresh and Huynh in J. Comput. Phys. 136(1):83-99, 1997, doi:10.1006/jcph.1997.5745), one of the most common high-order schemes currently employed in numerical-relativity simulations. We find that ELH achieves comparable and, in many of the cases studied here, better accuracy than more traditional methods at a fraction of the computational cost (up to \({\sim}50\%\) speedup). Given its accuracy and its simplicity of implementation, ELH is a promising framework for the development of new special- and general-relativistic hydrodynamics codes well adapted for massively parallel supercomputers.
Keywords
1 Introduction
Large-scale general-relativistic hydrodynamical numerical simulations have been shown to be a very powerful tool for the study of astrophysical systems involving compact objects such as black holes and neutron stars (Font 2008; Shibata and Taniguchi 2011; Rezzolla and Zanotti 2013; Martí and Müller 2015; Shibata 2016; Baiotti and Rezzolla 2016; Paschalidis 2016). The realisation of such simulations requires dealing with very different physical, mathematical and computational issues. One of the most challenging of such issues, and one that could lead to significant differences on the outcome of resolution-limited simulations, is the choice of the numerical method for the solution of the hydrodynamics equations (one such difference would be e.g., the dephasing of the gravitational waveforms in binary neutron star merger simulations, where different numerical schemes can lead to different dynamics of the matter bulk, see e.g., Baiotti et al. 2011; Read et al. 2013; Radice et al. 2014a; Thierfelder et al. 2011; Bernuzzi et al. 2012).
The most commonly used methods in this context are collectively known as high-resolution shock-capturing (HRSC) techniques (see Rezzolla and Zanotti 2013 for an introduction). Belonging to this class, which contains both finite-differences and finite-volumes schemes, are e.g., slope limiting methods (e.g., Roe 1985), the piece-wise parabolic method (PPM) (Colella and Woodward 1984; Martí and Müller 1996), the fifth-order monotonicity preserving (MP5) method (Suresh and Huynh 1997), essentially/weighted non-oscillatory (ENO/WENO) methods (Harten et al. 1987; Liu et al. 1994; Jiang and Shu 1996; De Pietri et al. 2016; Bernuzzi and Dietrich 2016) and many others.
HRSC methods are very effective in dealing with shocks and suppressing spurious oscillations, and have been employed with varying degree of success in astrophysical simulations. In recent times much work has gone into improving these schemes (e.g., by innovative mesh refinement techniques such as in DeBuhr et al. 2015) or moving beyond them; one promising alternative paradigm is that of discontinuous Galerkin methods (see, e.g., Radice and Rezzolla 2011; Bugner et al. 2016; Zanotti et al. 2015; Kidder et al. 2016; Miller and Schnetter 2016). Many of such schemes, however, potentially suffer from a few shortcomings: (i) they are complex to derive and implement, or to extend and modify (e.g., to increase the formal order of accuracy); (ii) they often depend on many coefficients that require some degree of optimisation (e.g., the WENO methods); (iii) they can lead to load imbalance in parallel implementations as a result of their complexity.
In this work we propose a different approach that is able to address some of these shortcomings (especially the points (i) and (iii) previously mentioned), which we refer to as ‘entropy-limited hydrodynamics’ (ELH) and formulate it in a finite-differences framework. The underlying concept is relatively straightforward: the hydrodynamical fluxes are computed using an unfiltered, high-order stencil, to which a contribution from a low-order, stable numerical flux (the Lax-Friedrichs flux) is added in order to ensure stability. To determine which gridpoints are in need of the low-order contribution, we employ a ‘shock detector’, which not only marks region of the computational domain requiring the limiter, but also determines the relative weights of the high and low-order fluxes.
The use of a hybrid numerical flux to achieve both accuracy and stability places our method in the class of flux-limiting schemes (see, e.g., the classification in Leveque 1992), which have long been a feature in the panorama of numerical schemes for hydrodynamics. In this context the Lax-Friedrichs flux is a common choice for low-order methods, being monotone and dissipative (a different example of combining high- and low-order methods, in the context of the reconstruction method, would be e.g., Tchekhovskoy et al. 2007).
To drive the activation of the Lax-Friedrichs method, a criterion to flag generically problematic points of the computational domain is needed. Such a criterion is offered by the entropy viscosity function described by Guermond et al. (we refer primarily to Guermond et al. 2011, but see also Guermond and Pasquetti 2008; Zingan et al. 2013), in which the local production of entropy is used to identify shocks. Since entropy is produced only in the presence of shocks, this results in a stable method able to recover high-order in regions of smooth flow. We extend the definition of the entropy viscosity from the classical to the relativistic case, and employ it to drive the flux limiting scheme rather than as a weight to additional viscous terms in the hydrodynamical equations. As a result, and in contrast to the approach by Guermond et al. (2011), we do not modify the underlying equations of relativistic hydrodynamics by introducing additional entropy-related terms.
In the following we describe the method and the details of our implementation, then report the results of tests we conducted in order to gauge its behaviour against a standard HRSC method, namely, MP5 (Suresh and Huynh 1997). The paper is structured as follows: in Section 2 we briefly summarise the equations of relativistic hydrodynamics and the finite-differences framework we employ. In Section 3 the ELH method and its implementation are described, while the results of the numerical tests are collected in Section 4. We present our conclusions and outlook in Section 5.
In the following we use the spacetime signature \((-,+,+,+)\), with Greek indices running from 0 to 3 and Latin indices from 1 to 3. We also employ the Einstein convention for the summation over repeated indices. Unless otherwise stated, all quantities are expressed in a geometrized system of units in which \(c=G=1\).
2 Relativistic hydrodynamics: theory and numerics overview
2.1 Relativistic hydrodynamics
We summarize here the mathematical framework of relativistic hydrodynamics. In the interest of simplicity, the discussion is limited to special relativity, while the general-relativistic case, which is relevant for the neutron-star tests of Section 4.2, can be found in Appendix 1.
Note that the source functions S are identically zero in special relativity, but this is no longer the case in a generic spacetime, where metric-dependent terms appear both in the fluxes and sources (see Appendix 1).
2.2 Numerical methods
The ELH method proposed here has been implemented in the code WhiskyTHCEL as a variant of the WhiskyTHC code (Radice and Rezzolla 2012; Radice et al. 2014a, 2014b) based on the Einstein Toolkit (Löffler et al. 2012; Zilhão and Löffler 2013; Einstein Toolkit 2010). WhiskyTHC implements both finite-difference and finite-volume methods applied to a characteristic-variables decomposition with a Lax-Friedrichs flux-splitting for upwinding. It also crucially provides a positivity preserving limiter to cope with large rest-mass density jumps, e.g., as those appearing across the surface of compact stars. In the following we summarise the main components of the underlying algorithm and refer the interested reader to Radice and Rezzolla (2012) and Radice et al. (2014a) for a more detailed description.
Furthermore, we select the MP5 scheme as our benchmark against which to test the properties of the EL method. MP5 is built on top of the U5 stencil, but the resulting fluxes are limited so as to preserve monotonicity near discontinuities (Suresh and Huynh 1997; Mignone et al. 2010; Radice and Rezzolla 2012). MP5 offers a good compromise between robustness and accuracy and it has been successfully employed in several realistic scenarios in which it also achieved high convergence-order (Radice et al. 2014a, 2014b; Radice et al. 2015). It has therefore become a de facto standard in our work, hence we use it as a reference.
The reconstruction procedure outlined above can be applied on each equation in the system (5) (this is also called a components split) or to its local characteristic variables (in which case it is referred to as a characteristics split).
We note that this algorithm does not free us from having to employ an artificial floor (or atmosphere) to treat (ideally) vacuum regions: these are filled with a uniform and tenuous fluid with rest-mass density \(\rho_{\mathrm{atmo}}\). Whenever in the subsequent evolution the rest-mass density of a gridpoint falls below the floor value \(\rho_{\mathrm{atmo}}\), it is reset to the floor value, its three-velocity is set to zero and the specific internal energy is set to the corresponding value coming from the EOS. In neutron star simulations we fix the floor at \(\rho_{\mathrm{atmo}}=10^{-16}~M_{\odot}^{-2}\), i.e. the typical value of \(\rho_{\mathrm{atmo}}/\rho_{\mathrm{max}}\) is ∼10^{−13}.
Details of the algorithms we employ to evolve the spacetime and couple it to the fluid evolution are given in Appendix 2.
The last step in the algorithm is the actual time evolution. Since after the spatial discretization, the original PDEs to be solved are in the form of a coupled systems of ODEs, this is taken care of in a method-of-lines (MOL) fashion by means of a fixed step Runge-Kutta time integrator. We employ either the standard fourth-order Runge-Kutta method (RK4) or a third-order (RK3, see Gottlieb et al. 2009) method with strong stability preserving (SSP) properties.
3 Entropy-limited hydrodynamics
3.1 Description of the scheme
The scheme we propose consists of two building blocks: (1) a function detecting shocks; (2) a limiter scheme of the high-order fluxes. We will start discussing the latter.
As customary in flux-limiting schemes, we modify the high-order approximation of the flux by combining (or ‘hybridising’) it with a (local) Lax-Friedrichs flux contribution as in (12).
The expected behaviour of the entropy residual is that it cannot decrease in time and that is spatially restricted to very small regions in the neighbourhood of shocks, ideally expressed a delta function peaked at the location \(\boldsymbol{x}_{s}\) of shocks, i.e., \(\mathcal{R} \propto \delta(\boldsymbol{x}-\boldsymbol{x}_{s})\). A physical justification for this latter expectation is rather simple to motivate. Euler equations generally apply to perfect fluids, and while they can capture non-ideal features (i.e., shocks), the description of the latter is only approximate. As long as the flow is smooth and the perfect-fluid approximation holds, all phenomena are reversible and there can be no production of entropy. However, in those regimes where the perfect-fluid approximation breaks down and non-ideal effects appear, namely, at the location of shocks, the entropy production is nonzero and the entropy jumps locally to a higher value. Since shocks are regions of dimension \(N-1\) in spatial manifolds with N spatial dimensions, the entropy residual \(\mathcal{R}\) must be a Dirac delta peaked at shock locations for it to provide a finite contribution.
An additional benefit of this definition is the ability of the resulting scheme in differentiating automatically between shocks and contact discontinuities. This follows from the fact that at contact discontinuities there is no entropy production and therefore the viscosity there would be zero as well (Rezzolla and Zanotti 2013).
3.2 Numerical implementation
A few remarks are useful at this point. First, the time derivative of the specific entropy in Eq. (21) is computed with a low-order method and this could in principle be a limiting factor for the convergence properties of the overall scheme. In practice, however, we find that the space discretization error dominates over the error on the time derivative in the tests we have performed, so that the scheme achieves high-order convergence as expected despite the use of a low-order approximation for \(\partial_{t} s\). Second, the high-order flux \(f^{\mathrm{HO}}\) is computed component by component. In fact since the reconstruction operators U5 and U7 ((9) and (10)) are linear, they commute with the matrices used to perform the characteristic decomposition, and there is therefore no difference in this case between component-by-component and characteristic decomposition. This contributes (along with other intrinsic differences in the formulation of the schemes) to a significant speed-up of the code (up to \({\sim}50\%\), depending on the setup of the grid on the computing nodes) with respect to MP5, since there is no need to compute the system eigenvectors and apply the resulting matrix. By contrast, the MP5 reconstruction is nonlinear and does not commute with the characteristic decomposition. As a result, when using MP5 we always switch to characteristic variables, since this is known to reduce spurious numerical oscillations in high-order methods (Suresh and Huynh 1997).
In addition to dampening oscillations in the viscosity mentioned in the previous section, the necessity of the smoothing step stems from the fact that the viscosity is computed once at the beginning of every new timestep before its value is used in (15). Since the viscosity is kept constant during the series of Runge-Kutta internal steps, it ‘lags behind’ in time with respect to the solution. This issue is addressed by the smoothing procedure, but in practice we have found that this does not represent a problem in our tests. The smoothing (24) also prevents the viscosity to plunge to very small values where it should instead be non negligible. This can happen, e.g., close to stellar surfaces as a result of oscillations in the solution. The application of the smoothing operator removes this problem by joining seamlessly the values of the viscosity in the neighbouring points.
4 Numerical tests
We report in this section the results of some of the tests obtained with the ELH method described in the previous sections. In all tests we compare the ELH results with those obtained using the monotonicity preserving, fifth-order scheme (MP5). In particular, unless otherwise stated, we couple the ELH method to the fifth-order U5 stencil (9), to make a fair and sensible comparison between methods of the same order. In few cases, however, we will also report results obtained with the seventh-order stencil U7 (10). We will refer to the corresponding schemes as to EL5 and EL7, respectively. Finally, it is useful to remark that in all of the following tests no attempt was made to tune the coefficients \(c_{e}\) and \(c_{\mathrm{max}}\) introduced in Section 3.1, and that have been set to unity here. Despite this very simple choice, the ELH method is stable and accurate in all cases considered, as the following sections make clear. At the same time, we consider it possible (if not likely) that the results could be further improved after a careful exploration of the changes in the solution upon a change of \(c_{e}\) and \(c_{\mathrm{max}}\); we will leave this exploration to a future work.
4.1 Special-relativistic tests
We begin with a series of mostly one-dimensional tests, performed in special-relativistic hydrodynamics, so that the metric \(g_{\mu\nu}\) is fixed to the flat Minkowski metric \(\eta_{\mu\nu}\) and no spacetime evolution is performed. Also, since we are mostly interested in the behaviour of the scheme in realistic astrophysical applications, we focus on just a handful of one-dimensional test cases: a smooth nonlinear wave and three shock-tube tests.
4.1.1 Smooth nonlinear wave
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.
4.1.2 Shock-tube tests
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.
4.2 Three-dimensional general-relativistic tests: neutron stars
We next test the EL5 scheme against a series of three-dimensional tests mostly based on the evolution of single, isolated neutron stars in general relativity (with the exception of grazing-collision test of Section 4.2.6). In each test we employ for the evolution the ideal-fluid EOS (4) with \(\Gamma=2\). The neutron star initial data is constructed using a polytropic EOS \(p=K\rho^{\tilde{\gamma}}\) also with \(\tilde{\gamma}=2\) and \(K=100~M_{\odot}^{-2}\).
4.2.1 Isolated star in the Cowling approximation
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.
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 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.
4.2.2 Isolated star in a dynamical spacetime
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.
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.
4.2.3 Perturbed isolated star
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.
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.
4.2.4 Migration test
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.
4.2.5 Isolated rotating neutron star
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.
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%.
4.2.6 Grazing collision of neutron stars
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 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.
4.2.7 Gravitational collapse to a black hole
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.
5 Conclusions
We have presented a new high-order numerical method for the solution of the Euler equations of general-relativistic hydrodynamics that we name ‘entropy-limited hydrodynamics’ (ELH). The scheme is of the flux-limiting type, where a high-order numerical flux is combined with a stable low-order method, namely the Lax-Friedrichs flux. The flux-limiting is activated and driven by a shock indicator based on a measure of the entropy generated by the solution. Such a special and general-relativistic method is inspired by the entropy-viscosity method proposed recently for the solution of the classical equations of hydrodynamics (Guermond et al. 2011), but it is also importantly different in that it does not require any change in the equations of relativistic hydrodynamics.
To assess the robustness and accuracy of our new method, we have discussed its implementation in the WhiskyTHCEL code, which exploits the finite-difference capabilities of the WhiskyTHC code, and tested its validity with an extensive series of tests, comparing the results of ELH with those obtained with another well-tested and high-order HRSC scheme: the fifth-order monotonicity-preserving MP5 method (Suresh and Huynh 1997). Overall, we have found that the scheme is stable and able to cope with shocks and discontinuities, both in classical test such as shock-tube tests, as well as in realistic astrophysical simulations.
Under all of these conditions the scheme has been found to be stable and to yield accuracy that is comparable, if not better, of that of the MP5 method. In some tests involving stars that are nonrotating or not moving across the grid, it also offers definite advantages, such as a sharper resolution of the surface/vacuum interface. At the same time, however, it also shows a less good conservation of the rest mass for stars that are rotating or moving across the computational domain (the opposite is true for stationary nonrotating stars, where the new method conserves rest mass more accurately). Quite surprisingly, all of the results presented here were obtained without any fine tuning of the two arbitrary coefficients that enter the definition of the scheme. Finally, thanks to its linearity and simplicity, the ELH method can also offer advantages in terms of performance. In our tests, we have found EL5 to be \({\sim}50\%\) faster than MP5, even though our implementation was not particularly optimized. For instance, a definite advantage of ELH, which we did not exploit, is that it can be easily vectorised. At the same time, we remark that the exact speed-up that can be achieved with ELH depends also on external factors, such as the grid setup and number of ghost zones, which may vary for different applications. An interesting development in this sense would be the use of this scheme in a discontinuous Galerkin framework, whose superior scalability properties should decouple the performance of the ELH method from the grid setup.
The work presented here could be improved in at least two ways. Firstly, the already good capturing properties of steep gradients as those given by the stellar surface, could be further enhanced and the full capabilities of the scheme further exploited by coupling it to truly multidimensional stencils. Second, the two free coefficients that appear in the method, and that we have here set to unity for simplicity, could potentially be tuned to optimise some of the features of the solution. Both of these aspects will be explored in future work.
In conclusion, we have shown that entropy-limited hydrodynamics is a robust, stable, and accurate alternative to commonly employed HRSC schemes. Its performance reaches the level of accuracy and stability necessary to apply it to realistic astrophysical simulations. Given these encouraging prospects, work is already in progress to apply this method to realistic simulations of binaries involving neutron stars and black holes.
Note that the operators (9) and (10) return approximations of the function h defined by \(f_{i}=:\int_{x_{i-1/2}}^{x_{i+1/2}} h(x') \, d x'\) at \(x_{i+1/2}\). In this sense, they act on volume averages, the point-wise flux being the volume average of h. The values \(h_{i\pm1/2}\) should appear in (8) instead of \(f_{i\pm1/2}\). We have here simplified the notation, but a full discussion can be found in Radice and Rezzolla (2012).
Further details can be found in Radice et al. (2014a). Here, θ is computed following the algorithm in Section 2.2 of Hu et al. (2013), but replacing the rest-mass density ρ with its conserved relativistic counterpart D.
The use of a higher-order stencil in the EL approach, e.g., EL7, does not yield to improvements in the solution; the treatment of the low-density regions is far more delicate and the mass conservation is degraded.
Of course, for both schemes the amount of rest-mass outside the star is minute, being only 10^{−7} of the initial rest-mass for the EL5 scheme and ∼10^{−5} for the MP5 scheme.
Declarations
Acknowledgements
We thank Kentaro Takami for providing the stellar oscillation eigenfrequencies, while Erik Schnetter, Ian Hinder, and Massimiliano Leoni for useful discussions. The simulations were performed on the SuperMUC cluster at the LRZ in Garching, on the LOEWE cluster in CSC in Frankfurt, on the HazelHen cluster at the HLRS in Stuttgart and on the Caltech compute cluster Zwicky.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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