An HLLC Riemann solver for relativistic flows – II. Magnetohydrodynamics

An approximate Riemann solver for the equations of relativistic magnetohydrodynamics (RMHD) is derived. The Harten–Lax–van Leer contact wave (HLLC) solver, originally developed by Toro, Spruce and Spears, generalizes the algorithm described in a previous paper to the case where magnetic fields are present. The solution to the Riemann problem is approximated by two constant states bounded by two fast shocks and separated by a tangential wave. The scheme is Jacobian-free, in the sense that it avoids the expensive characteristic decomposition of the RMHD equations and it improves over the HLL scheme by restoring the missing contact wave.
Multidimensional integration proceeds via the single step, corner transport upwind (CTU) method of Colella, combined with the constrained transport (CT) algorithm to preserve divergence-free magnetic fields. The resulting numerical scheme is simple to implement, efficient and suitable for a general equation of state. The robustness of the new algorithm is validated against one- and two-dimensional numerical test problems.     

An approximate Riemann solver for relativistic magnetohydrodynamics

A Godunov-type scheme for relativistic magnetohydrodynamic (MHD) equations is developed. We consider the Maxwell equations and dynamic equations for a gas with perfect conductivity in hyperbolic form as was suggested by van Putten. To calculate the fluxes of conservative variables through cells' interfaces we suggest an algorithm for the solution of the linearized Riemann problem. 'Primitive' variables are calculated by solving a non-linear system using the Newton method .     

A five-wave Harten–Lax–van Leer Riemann solver for relativistic magnetohydrodynamics

We present a five-wave Riemann solver for the equations of ideal relativistic magneto-hydrodynamics. Our solver can be regarded as a relativistic extension of the five-wave HLLD Riemann solver initially developed by Miyoshi & Kusano for the equations of ideal magnetohydrodynamics. The solution to the Riemann problem is approximated by a five-wave pattern, comprising two outermost fast shocks, two rotational discontinuities and a contact surface in the middle. The proposed scheme is considerably more elaborate than in the classical case since the normal velocity is no longer constant across the rotational modes. Still, proper closure to the Rankine–Hugoniot jump conditions can be attained by solving a non-linear scalar equation in the total pressure variable which, for the chosen configuration, has to be constant over the whole Riemann fan. The accuracy of the new Riemann solver is validated against one-dimensional tests and multidimensional applications. It is shown that our new solver considerably improves over the popular Harten–Lax–van Leer solver or the recently proposed HLLC schemes.     

An Efficient High-Resolution Shock-Capturing Scheme for Multi-Dimensional Flows I. Hydrodynamics

Many problems at the forefront of theoretical astrophysics require a treatment of dynamical fluid behavior. We present an efficient high-resolution shock-capturing hydrody-namic scheme designed to study such phenomena. We have implemented a weighted, essentially non-oscillatory (WENO) scheme to fifth order accuracy in space. HLLE approximate Riemann solver is used for the flux computation at cell interface, which does not require spectral decomposition into characteristic waves and so is computationally friendly. For time integration we apply a third order total variation diminishing (TVD) Runge-Kutta scheme. Extensive testing and comparison with schemes that require characteristic decomposition are carried out demonstrating the ability of our scheme to address challenging open questions in astrophysics.     

wham: a WENO-based general relativistic numerical scheme – I. Hydrodynamics

Active galactic nuclei, X-ray binaries, pulsars and gamma-ray bursts are all believed to be powered by compact objects surrounded by relativistic plasma flows driving phenomena such as accretion, winds and jets. These flows are often accurately modelled by the relativistic magnetohydrodynamic (MHD) approximation. Time-dependent numerical MHD simulations have proven to be especially insightful, but one regime that remains difficult to simulate is when the energy scales (kinetic, thermal, magnetic) within the plasma become disparate. We develop a numerical scheme that significantly improves the accuracy and robustness of the solution in this regime. We use a modified form of the weighted essentially non-oscillatory (WENO) method to construct a finite-volume general relativistic hydrodynamics code called wham that converges at fifth order. We avoid (1) field-by-field decomposition by adaptively reducing down to two-point stencils near discontinuities for a more accurate treatment of shocks and (2) excessive reduction to low-order stencils, as in the standard WENO formalism, by maintaining high-order accuracy in smooth monotonic flows. Our scheme performs the proper surface integral of the fluxes, converts cell-averaged conserved quantities to point-conserved quantities before performing the reconstruction step, and correctly averages all source terms. We demonstrate that the scheme is robust in strong shocks, very accurate in smooth flows and maintains accuracy even when the energy scales in the flow are highly disparate.     

Non-axisymmetric relativistic Bondi–Hoyle accretion on to a Schwarzschild black hole

We present the results of an exhaustive numerical study of fully relativistic non-axisymmetric Bondi–Hoyle accretion on to a moving Schwarzschild black hole. We have solved the equations of general relativistic hydrodynamics with a high-resolution shock-capturing numerical scheme based on a linearized Riemann solver. The numerical code was previously used to study axisymmetric flow configurations past a Schwarzschild black hole. We have analysed and discussed the flow morphology for a sample of asymptotically high Mach number models. The results of this work reveal that initially asymptotic uniform flows always accrete on to the hole in a stationary way, which closely resembles the previous axisymmetric patterns. This is in contrast with some Newtonian numerical studies where violent flip-flop instabilities were found. As discussed in the text, the reason can be found in the initial conditions used in the relativistic regime, as they cannot exactly duplicate the previous Newtonian setups where the instability appeared. The dependence of the final solution on the inner boundary condition as well as on the grid resolution has also been studied. Finally, we have computed the accretion rates of mass and linear and angular momentum.     

Modelling interaction of relativistic and non-relativistic winds in binary system PSR B1259−63/SS2883 – I. Hydrodynamical limit

In this paper, we present a detailed hydrodynamical study of the properties of the flow produced by the collision of a pulsar wind with the surrounding in a binary system. This work is the first attempt to simulate interaction of the ultrarelativistic flow (pulsar wind) with the non-relativistic stellar wind. Obtained results show that the wind collision could result in the formation of an 'unclosed' (at spatial scales comparable to the binary system size) pulsar wind termination shock even when the stellar wind ram pressure exceeds significantly the pulsar wind kinetic pressure. Moreover, the post-shock flow propagates in a rather narrow region, with very high bulk Lorentz factor (γ∼ 100). This flow acceleration is related to adiabatic losses which are purely hydrodynamical effects. Interestingly, in this particular case, no magnetic field is required for formation of the ultrarelativistic bulk outflow. The obtained results provide a new interpretation for the orbital variability of radio, X-ray and gamma-ray signals detected from binary pulsar system PSR B1259−63/SS2883.     

Hydrodynamical adaptive mesh refinement simulations of turbulent flows – I. Substructure in a wind

The problem of the resolution of turbulent flows in adaptive mesh refinement (AMR) simulations is investigated by means of three-dimensional (3D) hydrodynamical simulations in an idealized setup, representing a moving subcluster during a merger event. AMR simulations performed with the usual refinement criteria based on local gradients of selected variables do not properly resolve the production of turbulence downstream of the cluster. Therefore, we apply novel AMR criteria which are optimised to follow the evolution of a turbulent flow. We demonstrate that these criteria provide a better resolution of the flow past the subcluster, allowing us to follow the onset of the shear instability, the evolution of the turbulent wake and the subsequent back-reaction on the subcluster core morphology. We discuss some implications for the modelling of cluster cold fronts.     

3D models of radiatively driven colliding winds in massive O+O star binaries – I. Hydrodynamics

The dynamics of the wind–wind collision in massive stellar binaries are investigated using 3D hydrodynamical models which incorporate gravity, the driving of the winds, the orbital motion of the stars and radiative cooling of the shocked plasma. In this first paper, we restrict our study to main-sequence O+O binaries. The nature of the wind–wind collision region is highly dependent on the degree of cooling of the shocked plasma, and the ratio of the flow time-scale of the shocked plasma to the orbital time-scale. The pre-shock wind speeds are lower in close systems as the winds collide prior to their acceleration to terminal speeds. Radiative inhibition may also reduce the pre-shock wind speeds. Together, these effects can lead to rapid cooling of the post-shock gas. Radiative inhibition is less important in wider systems, where the winds are accelerated to higher speeds before they collide, and the resulting collision region can be largely adiabatic. In systems with eccentric orbits, cold gas formed during periastron passage can persist even at apastron, before being ablated and mixed into its surroundings and/or accelerated out of the system.     

Semiclassical approximation for low-degree stellar p modes – I. The classical eigenfrequency equation

A new eigenfrequency equation for low-degree solar-like oscillations in stars is developed, based on the assumption of purely classical propagation in the stellar interior of acoustic waves modified by buoyancy and gravity . Compared with high-frequency asymptotic analysis, the eigenfrequency equation has a new functional form, with expansion in powers of ℓ(ℓ+1) instead of 1/ ω . Basic observable quantities, the 'large' and 'small' frequency separations , are interpreted as the dependence on frequency and refraction angle of a classical action integral for waves propagating close to the stellar diameter. The new eigenfrequency equation gives a significant improvement in accuracy over previous analyses when tested with solar p modes, suggesting this as an alternative and more powerful tool for applications in stellar seismology.     

Variational principles for relativistic smoothed particle hydrodynamics

In this paper we show how the equations of motion for the smoothed particle hydrodynamics (SPH) method may be derived from a variational principle for both non-relativistic and relativistic motion when there is no dissipation. Because the SPH density is a function of the coordinates the derivation of the equations of motion through variational principles is simpler than in the continuum case where the density is defined through the continuity equation. In particular, the derivation of the general relativistic equations is more direct and simpler than that of Fock. The symmetry properties of the Lagrangian lead immediately to the familiar additive conservation laws of linear and angular momentum and energy. In addition, we show that there is an approximately conserved quantity which, in the continuum limit, is the circulation.     

In-shock cooling in numerical simulations

We model a one-dimensional shock-tube using smoothed particle hydrodynamics and investigate the consequences of having finite shock-width in numerical simulations caused by finite resolution of the codes. We investigate the cooling of gas during passage through the shock for three different cooling regimes.
For a theoretical shock temperature of 10⁵ K, the maximum temperature of the gas is much reduced. When the ratio of the cooling time to shock-crossing time was 8, we found a reduction of 25 per cent in the maximum temperature reached by the gas. When the ratio was reduced to 1.2, the maximum temperature reached dropped to 50 per cent of the theoretical value. In both cases the cooling time was reduced by a factor of 2.
At lower temperatures, we are especially interested in the production of molecular hydrogen, and so we follow the ionization level and H₂ abundance across the shock. The effect of in-shock cooling is substantial: the maximum temperature the gas reaches compared with the theoretical temperature is found to vary between 0.15 and 0.81, depending upon the shock strength and mass resolution. The downstream ionization level is reduced from the theoretical level by a factor of between 2.4 and 12.5, and the resulting H₂ abundance by a factor of 1.35 to 2.22.
At temperatures above 10⁵ K, radiative shocks are unstable and will oscillate. We find that the shock jump temperature varies by a factor of 20 because of these oscillations.
We conclude that extreme caution must be exercised when interpreting the results of simulations of galaxy formation.     

Hydrodynamical adaptive mesh refinement simulations of turbulent flows – II. Cosmological simulations of galaxy clusters

The development of turbulent gas flows in the intra-cluster medium and in the core of a galaxy cluster is studied by means of adaptive mesh refinement (AMR) cosmological simulations. A series of six runs was performed, employing identical simulation parameters but different criteria for triggering the mesh refinement. In particular, two different AMR strategies were followed, based on the regional variability of control variables of the flow and on the overdensity of subclumps, respectively. We show that both approaches, albeit with different results, are useful to get an improved resolution of the turbulent flow in the ICM. The vorticity is used as a diagnostic for turbulence, showing that the turbulent flow is not highly volume filling but has a large area-covering factor, in agreement with previous theoretical expectations. The measured turbulent velocity in the cluster core is larger than 200 km s⁻¹, and the level of turbulent pressure contribution to the cluster hydrostatic equilibrium is increased by using the improved AMR criteria.     

Galaxy formation and evolution – I. The Padua tree-sph code (pd-sph)

In this paper we report on pd-sph , the new tree-sph code developed in Padua. The main features of the code are described and the results of a new and independent series of 1D and 3D tests are shown. The paper is mainly dedicated to the presentation of the code and to the critical discussion of its performance. In particular, great attention is devoted to the convergency analysis. The code is highly adaptive in space and time by means of individual smoothing lengths and individual time-steps. At present it contains both dark and baryonic matter, this latter in the form of gas and stars, cooling, thermal conduction, star formation, feedback from Type I and II supernovae, stellar winds, and ultraviolet flux from massive stars, and finally chemical enrichment. New cooling rates that depend on the metal abundance of the interstellar medium are employed, and the differences with respect to the standard ones are outlined. Finally, we show the simulation of the dynamical and chemical evolution of a disc-like galaxy with and without feedback. The code is suitably designed to study in a global fashion the problem of formation and evolution of elliptical galaxies, and in particular to feed a spectrophotometric code from which the integrated spectra, magnitudes and colours (together with their spatial gradients) can be derived.