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Snapshots of the color-coded rest-mass density distributions of the 2D rotor problem at t = 0.4 using the code described in Antón et al. (2010). The results in the upper (bottom) panels were obtained with a grid of 1024² (128²) cells. The left, middle, and right panels show the results obtained with the FWD, HLLC, and HLL solver, respectively. Image reproduced with permission from Figure 14 of Antón et al. (2010), copyright by AAS.
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An overview of grid-based numerical methods used in relativistic hydrodynamics (RHD) and magnetohydrodynamics (RMHD) is presented. Special emphasis is put on a comprehensive review of the application of high-resolution shock-capturing methods. Results of a set of demanding test bench simulations obtained with different numerical methods are compare...
Citations
... For the relativistic plasma flow model based on RMHD equations, [42] introduced a Godunov type scheme. An overview of grid-based numerical methods used in relativistic hydrodynamics and magnetohydrodynamics is presented in [49]. A total variation diminishing scheme is designed in [7]. ...
This article proposes entropy stable discontinuous Galerkin schemes (DG) for two-fluid relativistic plasma flow equations. These equations couple the flow of relativistic fluids via electromagnetic quantities evolved using Maxwell's equations. The proposed schemes are based on the Gauss-Lobatto quadrature rule, which has the summation by parts (SBP) property. We exploit the structure of the equations having the flux with three independent parts coupled via nonlinear source terms. We design entropy stable DG schemes for each flux part, coupled with the fact that the source terms do not affect entropy, resulting in an entropy stable scheme for the complete system. The proposed schemes are then tested on various test problems in one and two dimensions to demonstrate their accuracy and stability. Keywords Entropy stable discontinuous Galerkin schemes · Two-fluid relativistic plasma flows · Balance laws Mathematics Subject Classification (2020) MSC 35L25 · MSC 35L56 · MSC 35L75 · MSC 65M60
... Various numerical methods have been developed for solving the RHD equations over the past few decades, including but not limited to finite difference methods [44,6,57,35,15,50], finite volume methods [28,42,1], discontinuous Galerkin (DG) methods [34,60,43,19], and so on. The interested readers are also referred to the review papers [11,25,26] for more related developments in this direction. ...
... Numerical codes are aimed at solving the RMHD equations (see Martí & Müller (2015) and for reviews). In our case, we use multi-dimensional high resolution shock-capturing methods, with a finite volume scheme. ...
In this paper, we review recent and ongoing work by our group on numerical simulations of relativistic jets. Relativistic outflows in astrophysics are related to dilute, high energy plasmas, with physical conditions out of the reach of current laboratory capabilities. Simulations are thus imperative for the study of these objects. We present a number of such scenarios that have been studied by our group at the Universitat de València. In particular, we have focused on the evolution of extragalactic outflows through galactic and intergalactic environments, deceleration by interaction with stars or clouds or the propagation of jets in X-ray binaries and interaction with stellar winds from massive companions. All also share their role as particle acceleration sites and production of non-thermal radiation throughout the electromagnetic spectrum. Therefore, our work is not only aimed at understanding the impact of outflows on their environments and thus their role in galaxy and cluster evolution, but also the nature and capabilities of these sites as generators of high- and very-high-energy radiation and cosmic rays.
... Various numerical methods have been developed for solving the RHD equations over the past few decades, including but not limited to finite difference methods [43,7,55,35,16,49], finite volume methods [29,41,1], discontinuous Galerkin (DG) methods [34,58,42,20], and so on. The interested readers are also referred to the review papers [12,26,27] for more related developments in this direction. ...
The relativistic hydrodynamics (RHD) equations have three crucial intrinsic physical constraints on the primitive variables: positivity of pressure and density, and subluminal fluid velocity. However, numerical simulations can violate these constraints, leading to nonphysical results or even simulation failure. Designing genuinely physical-constraint-preserving (PCP) schemes is very difficult, as the primitive variables cannot be explicitly reformulated using conservative variables due to relativistic effects. In this paper, we propose three efficient Newton--Raphson (NR) methods for robustly recovering primitive variables from conservative variables. Importantly, we rigorously prove that these NR methods are always convergent and PCP, meaning they preserve the physical constraints throughout the NR iterations. The discovery of these robust NR methods and their PCP convergence analyses are highly nontrivial and technical. As an application, we apply the proposed NR methods to design PCP finite volume Hermite weighted essentially non-oscillatory (HWENO) schemes for solving the RHD equations. Our PCP HWENO schemes incorporate high-order HWENO reconstruction, a PCP limiter, and strong-stability-preserving time discretization. We rigorously prove the PCP property of the fully discrete schemes using convex decomposition techniques. Moreover, we suggest the characteristic decomposition with rescaled eigenvectors and scale-invariant nonlinear weights to enhance the performance of the HWENO schemes in simulating large-scale RHD problems. Several demanding numerical tests are conducted to demonstrate the robustness, accuracy, and high resolution of the proposed PCP HWENO schemes and to validate the efficiency of our NR methods.
... These calculations model the relevant physics with different degrees of sophistication and employ different numerical schemes, details of which are beyond the scope of this text (see e.g. Baumgarte & Shapiro, 2010;Rezzolla & Zanotti, 2013;Martí & Müller, 2015;Rosswog, 2015;Duez & Zlochower, 2019). Even by the aid of supercomputers it is challenging to achieve high numerical resolution to resolve for instance the steep density gradients of NSs, turbulent matter motion or the detailed structure and evolution of mass outflows, which typically expand away from the central object, i.e. into regions of coarser numerical resolution. ...
Neutron stars (NSs) and black holes (BHs) are born when the final collapse of the stellar core terminates the lives of stars more massive than about 9 Msun. This can trigger the powerful ejection of a large fraction of the star's material in a core-collapse supernova (CCSN), whose extreme luminosity is energized by the decay of radioactive isotopes such as 56Ni and 56Co. When evolving in close binary systems, the compact relics of such infernal catastrophes spiral towards each other on orbits gradually decaying by gravitational-wave emission. Ultimately, the violent collision of the two components forms a more massive, rapidly spinning remnant, again accompanied by the ejection of considerable amounts of matter. These merger events can be observed by high-energy bursts of gamma rays with afterglows and electromagnetic transients called kilonovae, which radiate the energy released in radioactive decays of freshly assembled rapid neutron-capture elements. By means of their mass ejection and the nuclear and neutrino reactions taking place in the ejecta, both CCSNe and compact object mergers (COMs) are prominent sites of heavy-element nucleosynthesis and play a central role in the cosmic cycle of matter and the chemical enrichment history of galaxies. The nuclear equation of state (EoS) of NS matter, from neutron-rich to proton-dominated conditions and with temperatures ranging from about zero to ~100 MeV, is a crucial ingredient in these astrophysical phenomena. It determines their dynamical processes, their remnant properties even at the level of deciding between NS or BH, and the properties of the associated emission of neutrinos, whose interactions govern the thermodynamic conditions and the neutron-to-proton ratio for nucleosynthesis reactions in the innermost ejecta. This chapter discusses corresponding EoS dependent effects of relevance in CCSNe as well as COMs. (slightly abridged)
... There exist several approaches to numerically treat relativistic hydrodynamics: most prominent are Eulerian grid-based methods (including finitedifference, finite-volume or discontinuous Galerkin schemes) and Lagrangian smoothed particle hydrodynamics (SPH) (for reviews see e.g. Wilson & Mathews 2003;Font 2008;Alcubierre 2008;Baumgarte & Shapiro 2010;Rezzolla & Zanotti 2013;Rosswog 2015;Martí & Müller 2015;Shibata 2015, and references therein). Font (2008), Baiotti & Rezzolla (2017) and Foucart et al. (2022) provide a survey of codes currently used to tackle general relativistic hydrodynamics (GRHD) problems mostly in the context of binary mergers, and Liptai & Price (2019); Rosswog & Diener (2021) present some recent relativistic SPH tools. ...
We implement general relativistic hydrodynamics in the moving-mesh code AREPO. We also couple a solver for the Einstein field equations employing the conformal flatness approximation. The implementation is validated by evolving isolated static neutron stars using a fixed metric or a dynamical spacetime. In both tests the frequencies of the radial oscillation mode match those of independent calculations. We run the first moving-mesh simulation of a neutron star merger. The simulation includes a scheme to adaptively refine or derefine cells and thereby adjusting the local resolution dynamically. The general dynamics are in agreement with independent smoothed particle hydrodynamics and static-mesh simulations of neutron star mergers. Coarsely comparing, we find that dynamical features like the post-merger double-core structure or the quasi-radial oscillation mode persist on longer time scales, likely reflecting a lower numerical diffusivity of our method. Similarly, the post-merger gravitational wave emission shows the same features as observed in simulations with other codes. In particular, the main frequency of the post-merger phase is found to be in good agreement with independent results for the same binary system, while, in comparison, the amplitude of the post-merger gravitational wave signal falls off slower, i.e. the post-merger oscillations are less damped. The successful implementation of general relativistic hydrodynamics in the moving-mesh AREPO code, including a dynamical spacetime evolution, provides a fundamentally new tool to simulate general relativistic problems in astrophysics.
... Relativistic second-order fluid dynamics has become an essential tool in the description of the space-time evolution of high-energy phenomena, ranging from astrophysical systems like accretion flows [1], stellar collapse, gamma-ray bursts, and relativistic jets [2][3][4][5], to cosmology [6] and relativistic nuclear collisions at BNL-RHIC and CERN-LHC [7][8][9][10][11][12]. The space-time evolution of such systems and the interactions among their constituents are characterized not only in terms of an equation of state, but also by non-equilibrium transport processes. ...
We derive the transport coefficients of second-order fluid dynamics with $14$ dynamical moments using the method of moments and the Chapman-Enskog method in the relaxation-time approximation for the collision integral of the relativistic Boltzmann equation. Contrary to results previously reported in the literature, we find that the second-order transport coefficients derived using the two methods are in perfect agreement. Furthermore, we show that, unlike in the case of binary hard-sphere interactions, the diffusion-shear coupling coefficients $\ell_{V\pi}$, $\lambda_{V\pi}$, and $\tau_{V\pi}$ actually diverge in some approximations when the expansion order $N_\ell \rightarrow \infty$. Here we show how to circumvent such a problem in multiple ways, recovering the correct transport coefficients of second-order fluid dynamics with $14$ dynamical moments. We also validate our results for the diffusion-shear coupling by comparison to a numerical solution of the Boltzmann equation for the propagation of sound waves in an ultrarelativistic ideal gas.
... Nunez-de la Rosa & Munz (2016). The eigenvalues for the system of conservation laws given in Eq. (9) in the -direction are (Martí & Müller 2015), ...
... The numerical solver used in the present research is built upon a finite volume Godunov-type High-Resolution Shock-Capturing (HRSC) framework. This general methodology is extensively applied to RHD problems throughout the literature, recent develop-ments are presented in an excellent review by Martí & Müller Martí & Müller (2015). ...
This work assesses the dissipative properties of high-order numerical methods for relativistic hydrodynamics (RHD). A causal theory of physical dissipation is included within a finite volume High-Resolution Shock-Capturing (HRSC) framework based on the Israel-Stewart theory to study high-order WENO (Weighted-Essentially Non-Oscillatory) schemes for simulating the relativistic Kelvin-Helmholtz instability. We provide an estimation of the numerical dissipation of high-order schemes based on results obtained both with and without physically resolved dissipation and determine an empirical relationship between the numerical dissipation and the grid resolution. We consider the appearance of secondary flow features within the evolution of the Kelvin–Helmholtz instability and determine that they are numerical artifacts-this is partly based on arguments presented in terms of a frame-dependent form of the relativistic Reynolds number. There is a potential advantage of using high-order schemes in terms of their accuracy and computational cost on coarser grid resolutions when directly compared to low-order schemes on a fine grid in the presence of physical viscosity. It is possible to find reasonable agreement between numerical results that employ lower-order schemes using a finer grid resolution and results that employ higher order schemes at a coarser grid resolution when sufficient viscosity is present. Overall, the present analysis gives an insight into the numerical dissipation of high-order shock-wave capturing schemes which can be relevant to computational studies of astrophysical phenomena in the relativistic regime. The results presented herein are problem and scheme-dependent and serve to highlight the different roles of numerical and physical dissipation.
... This test is a relativistic version of "Sod's shocktube" [64], which has become a standard benchmark for relativistic hydrodynamics codes [65][66][67][68][69][70]. Apart from demonstrating that our code solves the relativistic hydrodynamics equations correctly, we used this test here also to show where our steering method (see Section 2.1.3) ...
The observation of gravitational waves from compact objects has now become an active part of observational astronomy. For a sound interpretation, one needs to compare such observations against detailed Numerical Relativity simulations, which are essential tools to explore the dynamics and physics of compact binary mergers. To date, essentially all simulation codes that solve the full set of Einstein’s equations are performed in the framework of Eulerian hydrodynamics. The exception is our recently developed Numerical Relativity code SPHINCS_BSSN which solves the commonly used BSSN formulation of the Einstein equations on a structured mesh and the matter equations via Lagrangian particles. We show here, for the first time, SPHINCS_BSSN neutron star merger simulations with piecewise polytropic approximations to four nuclear matter equations of state. In this set of neutron star merger simulations, we focus on perfectly symmetric binary systems that are irrotational and have 1.3 M⊙ masses. We introduce some further methodological refinements (a new way of steering dissipation, an improved particle–mesh mapping), and we explore the impact of the exponent that enters in the calculation of the thermal pressure contribution. We find that it leaves a noticeable imprint on the gravitational wave amplitude (calculated via both quadrupole approximation and the Ψ4 formalism) and has a noticeable impact on the amount of dynamic ejecta. Consistent with earlier findings, we only find a few times 10−3M⊙ as dynamic ejecta in the studied equal mass binary systems, with softer equations of state (which are more prone to shock formation) ejecting larger amounts of matter. In all of the cases, we see a credible high-velocity (∼0.5…0.7c) ejecta component of ∼10−4M⊙ that is launched at contact from the interface between the two neutron stars. Such a high-velocity component has been suggested to produce an early, blue precursor to the main kilonova emission, and it could also potentially cause a kilonova afterglow.
... This test is a relativistic version of "Sod's shocktube" [64] which has become a standard benchmark for relativistic hydrodynamics codes [65][66][67][68][69][70]. Apart from demonstrating that our code solves the relativistic hydrodynamics equations correctly, we use this test here also to show where our steering method, see Sec. 2.1.3, ...
The observation of gravitational waves from compact objects has now become an active part of observational astronomy. For a sound interpretation, one needs to compare such observations against detailed Numerical Relativity simulations, which are essential tools to explore the dynamics and physics of compact binary mergers. To date, essentially all simulation codes that solve the full set of Einstein's equations are performed in the framework of Eulerian hydrodynamics. The exception is our recently developed Numerical Relativity code \SpB which solves the commonly used BSSN formulation of Einstein equations on a structured mesh and the matter equations via Lagrangian particles. We show here, for the first time, \SpB neutron star merger simulations with piecewise polytropic approximations to four nuclear matter equations of state. We introduce some further methodological refinements (a new way of steering dissipation, an improved particle-mesh mapping) and we explore the impact of the exponent that enters in the calculation of the thermal pressure contribution. We find that it leaves a noticeable imprint on the gravitational wave amplitude (calculated via both quadrupole approximation and the $\Psi_4$-formalism) and has a noticeable impact on the amount of dynamic ejecta. Consistent with earlier findings, we only find a few times $10^{-3}$ \Msun as dynamic ejecta in the studied equal mass binary systems, with softer equations of state (which are more prone to shock formation) ejecting larger amounts of matter. We also see a credible high-velocity ($\sim0.5 .. 0.7c$) ejecta component of $\sim 10^{-4}$ \Msun in all our cases. Such a high-velocity component has been suggested to produce an early, blue precursor to the main kilonova emission and it could also potentially cause a kilonova afterglow.
