Relativistic Hydrodynamic Equations Research Articles

Efficient Solver of Relativistic Hydrodynamics with an Implicit Runge–Kutta Method

Abstract We propose a new method to solve the relativistic hydrodynamic equations based on implicit Runge–Kutta methods with a locally optimized fixed-point iterative solver. For numerical demonstration, we implement our idea for ideal hydrodynamics using the one-stage Gauss–Legendre method as an implicit method. The accuracy and computational cost of our new method are compared with those of explicit ones for the (1+1)D Riemann problem, as well as the (2+1)D Gubser flow and event-by-event initial conditions for heavy-ion collisions generated by TRENTo. We demonstrate that the solver converges with only one iteration in most cases, and as a result, the implicit method requires a smaller computational cost than the explicit one at the same accuracy in these cases, while it may not converge with an unrealistically large Δt. By showing a relationship between the one-stage Gauss–Legendre method with the iterative solver and the two-step Adams–Bashforth method, we argue that our method benefits from both the stability of the former and the efficiency of the latter.

Gradient-annihilated PINNs for solving Riemann problems: Application to relativistic hydrodynamics

We present a novel methodology based on Physics-Informed Neural Networks (PINNs) for solving systems of partial differential equations admitting discontinuous solutions. Our method, called Gradient-Annihilated PINNs (GA-PINNs), introduces a modified loss function that forces the model to partially ignore high-gradients in the physical variables, achieved by introducing a suitable weighting function. The method relies on a set of hyperparameters that control how gradients are treated in the physical loss. The performance of our methodology is demonstrated by solving Riemann problems in special relativistic hydrodynamics, extending earlier studies with PINNs in the context of the classical Euler equations. The solutions obtained with the GA-PINN model correctly describe the propagation speeds of discontinuities and sharply capture the associated jumps. We use the relative l2 error to compare our results with the exact solution of special relativistic Riemann problems, used as the reference “ground truth”, and with the corresponding error obtained with a second-order, central, shock-capturing scheme. In all problems investigated, the accuracy reached by the GA-PINN model is comparable to that obtained with a shock-capturing scheme, achieving a performance superior to that of the baseline PINN algorithm in general. An additional benefit worth stressing is that our PINN-based approach sidesteps the costly recovery of the primitive variables from the state vector of conserved variables, a well-known drawback of grid-based solutions of the relativistic hydrodynamics equations. Due to its inherent generality and its ability to handle steep gradients, the GA-PINN methodology discussed in this paper could be a valuable tool to model relativistic flows in astrophysics and particle physics, characterized by the prevalence of discontinuous solutions.

Provably convergent Newton–Raphson methods for recovering primitive variables with applications to physical-constraint-preserving Hermite WENO schemes for relativistic hydrodynamics

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. Our NR methods are versatile and can be seamlessly incorporated into any RHD schemes that require the recovery of primitive variables. As an application, we apply them 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.

Three‐dimensional stationary supersonic flows with axial symmetry in relativistic hydrodynamics

AbstractThe study of supersonic flows in relativistic hydrodynamics (RHDs) is of some interest for cosmology. This paper studies two types of three‐dimensional (3D) stationary supersonic flows with axial symmetry in RHDs: supersonic flows expanding in vacuum and supersonic flows against an infinite curved cone. To study supersonic flows expanding in vacuum, we consider a Cauchy problem for the 3D steady RHD equations with initial data that are a combination of an axisymmetric supersonic flow in a circular domain and vacuum in the remaining domain. Global existence of a piecewise smooth solution expanding in vacuum to the Cauchy problem is obtained, provided that the initial mass‐energy density is sufficiently small. When a supersonic flow arrives and hits a curved cone, a shock wave will arise around the cone. Moreover, if the vertex angle of the cone is less than a critical angle, then the shock will be attached to the vertex of the cone. We obtain the global existence of attached conical shock waves, provided that the curved cone satisfies some suitable conditions and the speed of the incoming flow is close to a limit speed.

Evolution of Nonlinear Wave Pulses in the sine-Gordon Equation Theory

We give the solution of the Whitham modulation equations for envelopes of pulses evolving according to the sine-Gordon equation. The Whitham equations are interpreted as the equations of relativistic hydrodynamics and their solving is reduced by the hodograph method to solving a linear partial differential equation. We describe the class of solutions of this equation with separation of variables and illustrate the theory by an example of the nonlinear wave packet evolution accompanied by its shrinking and decrease of the number of oscillations in the Whitham nonlinear region.

A Piecewise-Parabolic Reconstruction of the Physical Variables in a Low-Dissipation HLL Method for the Numerical Solution of the Equations of Special Relativistic Hydrodynamics

A Piecewise-Parabolic Reconstruction of the Physical Variables in a Low-Dissipation HLL Method for the Numerical Solution of the Equations of Special Relativistic Hydrodynamics

A hydrodynamical description of gravitational waves

It is easy to reason that gravity might be the effect of a fluid in disguise, as it will naturally arise in emergent gravity models where gravity is due to the effect of some fundamental particles, with the latter expected to behave collectively like a fluid at the macroscopic scale. We call this the fluid/gravity equivalence. The key difficulty with the fluid/gravity equivalence is to find the correct metric–fluid relation (the relation between the emergent metric and the fluid properties) so that the fluid not only has physically acceptable properties but also obeys the usual hydrodynamic equations, while at the same time the emergent metric also obeys the Einstein equations. Faced with the problem, we have previously made a tentative proposal of the metric–fluid relation, focusing only on obtaining physically acceptable predictions on the fluid properties. In this paper, however, we find that for the general gravitational wave spacetime near the null infinity, the underlying fluid not only has physically acceptable properties, but also satisfies the expected relativistic hydrodynamic equations in the Minkowski background, thus providing a concrete example satisfying both of the major requirements expected for the fluid/gravity equivalence.

К построению релятивистских гидродинамических уравнений в искривленном пространстве-времени, записанных в абстрактных геометрических обозначениях

This paper presents a macroscopic derivation of the system of equations of relativistic dissipative hydrodynamics in an abstract geometrical form, which does not depend on the chosen space-time coordinates and reduces to equations valid in both flat and curved space-time. The above procedure, serving as a link between the non-gravitational laws of physics and gravitation, is based on Einstein's equivalence principle which guarantees that the laws of physics written in abstract geometrical representation have the same form both in flat and curved spacetime. A modern formulation of relativistic hydrodynamics based on both classical nonequilibrium thermodynamics (first-order theory) and expanded irreversible thermodynamics (second-order theory) is proposed. The theory presented has both purely conceptual and applied significance. In particular, this theory has applications in such important fields of knowledge as nuclear physics, astrophysics and cosmology. For example, in viscous cosmological models, bulk viscosity acts as a cause of dissipation, which has a significant impact on the processes in the Universe.

Конструирование релятивистской гидродинамики многокомпонентной жидкости. 1. Метод релятивистской необратимой термодинамики

The paper develops relativistic mechanics and irreversible thermodynamics of a cosmological liquid mixture in which thermal conduction, diffusion, viscous flow and their crossing phenomena can occur. The main thermodynamic fields that occur in relativistic irreversible thermodynamics are defined as statistical expressions using relativistic kinetics. The effects of the Eckart approach on the choice of the hydrodynamic velocity are shown. The equations of relativistic multicomponent hydrodynamics for local densities of momentum, energy and number of particles of different sort, are also obtained from the fundamental conservation laws of relativistic thermodynamics.. The energy conservation law (the first beginning of relativistic thermodynamics) is formulated. Covariant Gibbs relation and local form of the second principle of thermodynamics in the presence of entropy source were derived in order to obtain determining relations linearly connecting fluxes with corresponding thermodynamic forces. Equivalent forms of relativistic entropy production in the form of a bilinear expression in terms of thermodynamic forces were obtained. A new cross-effect between diffusion and conduction arising from the relativistic term in the thermodynamic force coupled to the heat flux is discussed. The synopsis presented is of interest for relativistic mechanics, astrophysics, and cosmology.

Relativistic spin hydrodynamics

In recent years, due to the discoveries of hyperon spin polarization and vector meson spin alignment in relativistic heavy-ion collision experiments, the spin transports in quark-gluon matter has received intensive studies. The relativistic spin hydrodynamics is one of the important theoretical frameworks to describe the spin transports, which encodes the spin degree of freedom into a hydrodynamic theory. The relativistic spin hydrodynamics have the conservation equations for energy-momentum tensor, currents and total angular momentum. In this article, we give an overview of the recent progresses of the relativistic spin hydrodynamics. We focus on the following topics: 1) The derivation of the relativistic spin hydrodynamic equations, including the phenomenological approach, the effective theory method, and the kinetic approach, 2) Some special properties of spin hydrodynamics, especially the asymmetric energy-momentum tensor and the pseudogauge transformation, and 3) The analytical solutions to the relativistic spin hydrodynamics for systems under Bjorken and Gubser expansion.

Using piecewise-parabolic reconstruction of physics variables to constructing a low-dissipation HLL method for numerical solution of special relativistic hydrodynamics equations

Using piecewise-parabolic reconstruction of physics variables to constructing a low-dissipation HLL method for numerical solution of special relativistic hydrodynamics equations

To the derivation of relativistic hydrodynamic equations for a rarefied non-ideal gas system of high-energy particles in the framework of Tsallis statistics

The paper discusses the construction of non-extensive relativistic dissipative hydrodynamics of an anomalous hadronic fluid on the basis of relativistic kinetic equation, obtained earlier in the context of the Tsallis statistics, characterized by the nonextensivity parameter q, and taking into account correlation effects (by rejecting the standard hypothesis of molecular chaos) in the collision term for heavy ions. It is shown that some specific form of local thermal equilibrium quark-gluon matter is described by a generalized version of the relativistic Yüttner distribution. With the help of this distribution all thermodynamic parameters of state are defined in explicit form. Linear constitutive relations and transport coefficients such as shear viscosity, bulk viscosity and heat conductivity are derived from the linearized collision integral written in the Anderson-Witting form and evaluated using a relaxation time approximation. The designed non-extensive relativistic fluid dynamics is designed to simulate a wide range of phenomena in astrophysics, cosmology and high-energy physics.

Multicomponent second-order dissipative relativistic hydrodynamics with binary reactive collisions

We derive the multi-component second-order dissipative relativistic hydrodynamic equations using the moment-expansion method. By computing the transport coefficients using hard-sphere interactions, we investigate the role of multiple components and the reactive collisions. We find that both of these factors increase the effective cross-section and hence decrease the transport coefficients and relaxation times. We further compute such transport properties using leading-order perturbative QCD cross-sections. For both types of cross-sections, we find that the ratio between vector current relaxation time and conductivity for a multi-component fluid is notably different from that for a single-component fluid. Therefore, the current study provides a more applicable guideline for such a ratio in phenomenological hydrodynamics simulations.

Explosion of a Minimum-Mass Neutron Star within Relativistic Hydrodynamics

The relativistic hydrodynamics equations are adapted for the spherically symmetric case and the Lagrangian form. They are used to model the explosive disruption of a minimum-mass neutron star: a key ingredient of the stripping model for short gamma-ray bursts. The shock breakout from the neutron star surface accompanied by the acceleration of matter to ultrarelativistic velocities is studied. A comparison with the results of previously published nonrelativistic calculations is made.

Electron-acoustic solitons in dense electron–positron–ion plasma: Degenerate relativistic enthalpy function

An enthalpy function Hj∝1+ηj2 is deduced from the density of states for degenerate relativistic charged particulate. Here ηj=pj/mjc2 stands for the relativistic factor, pjmj is the Fermi momentum (mass) for the jth degenerate relativistic species and c designates the vacuum speed of light. The enthalpy function reduces the results Weinberg (1972), Lee and Cheong (2007) obtained for the nondegenerate nonrelativistic (NR) and ultrarelativistic (UR) fluids. The linear and nonlinear propagation characteristics of the electron-acoustic (EA) solitons are examined for dense electron–positron–ion (EPI) plasma that comprises degenerate relativistic electrons/positrons and non-degenerate ions. By using Hj and with the aid of the relativistic hydrodynamic equations, a linear dispersion relation for the EA waves is derived that admits the evolution of the fast and slow electron-acoustic (FEA/SEA) excitations. For the nonlinear EA waves, a Korteweg–de Vries (KdV) equation is obtained; numerical analysis reveals that the FEA mode may ensue in the solitary potential. The propagation of the SEA solitons is forbidden as it suffers from a negative phase dispersion effect. At a relatively low concentration of positrons, the EPI plasma may admit the evolution of electron–holes and therefore, give rise to the compressive FEA solitons. This investigation is important for understanding localized excitations in degenerate relativistic plasmas, as that are relevant in white dwarfs, neutron stars, and high-energy density facilities.

Entropy stable discontinuous Galerkin schemes for the special relativistic hydrodynamics equations

This article presents entropy stable discontinuous Galerkin numerical schemes for equations of special relativistic hydrodynamics with the ideal equation of state. The numerical schemes use the summation by parts (SBP) property of the Gauss-Lobatto quadrature rules. To achieve entropy stability for the scheme, we use two-point entropy conservative numerical flux inside the cells and a suitable entropy stable numerical flux at the cell interfaces. The resulting semi-discrete scheme is then shown to be entropy stable. Time discretization is performed using SSP Runge-Kutta methods. Several numerical test cases are presented to validate the accuracy and stability of the proposed schemes.

Family of new exact solutions for longitudinally expanding ideal fluids

We report on the discovery of new analytical solutions of the equations of relativistic ideal hydrodynamics. In this solution, the fluid expands in the longitudinal direction and contains a plateau structure that extends over a finite range in rapidity and can be either symmetric or asymmetric in that variable. We further calculate the corresponding pseudo-rapidity distribution of hadron yields, and find decent agreement with experimental measurements in high-energy Pb+Pb, Au+Au, p+Pb, and d+Au collisions.

Fate of twin stars on the unstable branch: Implications for the formation of twin stars

Hybrid hadron-quark equations of state that give rise a third family of stable compact stars have been shown to be compatible with the LIGO-Virgo event GW170817. Stable configurations in the third family are called hybrid hadron-quark stars. The equilibrium stable hybrid hadron-quark star branch is separated by the stable neutron star branch with a branch of unstable hybrid hadron-quark stars. The end-state of these unstable configurations has not been studied, yet, and it could have implications for the formation and existence of twin stars -- hybrid stars with the same mass as neutron stars but different radii. We modify existing hybrid hadron-quark equations of state with a first-order phase transition in order to guarantee a well-posed initial value problem of the equations of general relativistic hydrodynamics, and study the dynamics of non-rotating or rotating unstable twin stars via 3-dimensional simulations in full general relativity. We find that unstable twin stars naturally migrate toward the hadronic branch. Before settling into the hadronic regime, these stars undergo (quasi)radial oscillations on a dynamical timescale while the core bounces between the two phases. Our study suggests that it may be difficult to form stable twin stars if the phase transition is sustained over a large jump in energy density, and hence it may be more likely that astrophysical hybrid hadron-quark stars have masses above the twin star regime. We also study the minimum-mass instability for hybrid stars, and find that these configurations do not explode, unlike the minimum-mass instability for neutron stars. Additionally, our results suggest that oscillations between the two Quantum Chromodynamic phases could provide gravitational wave signals associated with such phase transitions in core-collapse supernovae and white dwarf-neutron star mergers.

High-order accurate entropy stable adaptive moving mesh finite difference schemes for special relativistic (magneto)hydrodynamics

This paper develops high-order accurate entropy stable (ES) adaptive moving mesh finite difference schemes for the two- and three-dimensional special relativistic hydrodynamic (RHD) and magnetohydrodynamic (RMHD) equations, which is the high-order accurate extension of Duan and Tang (2021) [23]. The key point is the derivation of the higher-order accurate entropy conservative (EC) and ES finite difference schemes in the curvilinear coordinates by carefully dealing with the discretization of the temporal and spatial metrics and the Jacobian of the coordinate transformation and constructing the high-order EC and ES fluxes with the discrete metrics. The spatial derivatives in the source terms of the symmetrizable RMHD equations and the geometric conservation laws are discretized by using the linear combinations of the corresponding second-order case to obtain high-order accuracy. Based on the proposed high-order accurate EC schemes and the high-order accurate dissipation terms built on the WENO reconstruction, the high-order accurate ES schemes are obtained for the RHD and RMHD equations in the curvilinear coordinates. The mesh iteration redistribution or adaptive moving mesh strategy is built on the minimization of the mesh adaption functional. Several numerical tests are conducted to validate the shock-capturing ability and high efficiency of our high-order accurate ES adaptive moving mesh methods on the parallel computer system with the MPI communication. The numerical results show that the high-order accurate ES adaptive moving mesh schemes outperform both their counterparts on the uniform mesh and the second-order ES adaptive moving mesh schemes.

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