We present an extension of relativistic single-particle distribution function for weakly interacting particles at local thermodynamical equilibrium including spin degrees of freedom, for massive spin 1/2 particles. We infer, on the basis of the global equilibrium case, that at local thermodynamical equilibrium particles acquire a net polarization proportional to the vorticity of the inverse temperature four-vector field. The obtained formula for polarization also implies that a steady gradient of temperature entails a polarization orthogonal to particle momentum. The single-particle distribution function in momentum space extends the so-called Cooper-Frye formula to particles with spin 1/2 and allows to predict their polarization in relativistic heavy ion collisions at the freeze-out.

Aims. We present a new numerical code, ECHO, based on a Eulerian conservative high-order scheme for time dependent threedimensional general relativistic magnetohydrodynamics (GRMHD) and magnetodynamics (GRMD). ECHO is aimed at providing a shock-capturing conservative method able to work at an arbitrary level of formal accuracy (for smooth flows), where the other existing GRMHD and GRMD schemes yield an overall second order at most. Moreover, our goal is to present a general framework based on the 3 + 1 Eulerian formalism, allowing for different sets of equations and different algorithms and working in a generic space-time metric, so that ECHO may be easily coupled to any solver for Einstein's equations. Methods. Our finite-difference conservative scheme previously developed for special relativistic hydrodynamics and MHD is extended here to the general relativistic case. Various high-order reconstruction methods are implemented and a two-wave approximate Riemann solver is used. The induction equation is treated by adopting the upwind constrained transport (UCT) procedures, appropriate to preserving the divergence-free condition of the magnetic field in shock-capturing methods. The limiting case of magnetodynamics (also known as force-free degenerate electrodynamics) is implemented by simply replacing the fluid velocity with the electromagnetic drift velocity and by neglecting the contribution of matter to the stress tensor. Results. ECHO is particularly accurate, efficient, versatile, and robust. It has been tested against several astrophysical applications, like magnetized accretion onto black holes and constant angular momentum thick disks threaded by toroidal fields. A novel test of the propagation of large-amplitude, circularly polarized Alfvén waves is proposed, and this allows us to prove the spatial and temporal high-order properties of ECHO very accurately. In particular, we show that reconstruction based on a monotonicity-preserving (MP) filter applied to a fixed 5-point stencil gives highly accurate results for smooth solutions, both in flat and curved metric (up to the nominal fifth order), while at the same time providing sharp profiles in tests involving discontinuities.

Conservative numerical schemes for general relativistic magnetohydrodynamics (GRMHD) require a method for transforming between "conserved" variables such as momentum and energy density and "primitive" variables such as rest-mass density, internal energy, and components of the four-velocity. The forward transformation (primitive to conserved) has a closed-form solution, but the inverse transformation (conserved to primitive) requires the solution of a set of five nonlinear equations. Here we discuss the mathematical properties of the inverse transformation and present six numerical methods for performing the inversion. The first method solves the full set of five nonlinear equations directly using a Newton-Raphson scheme and a guess from the previous

Abstract.A third order shock-capturing numerical scheme for three-dimensional special relativistic magnetohydrodynamics (3-D RMHD) is presented and validated against several numerical tests. The simple and efficient central scheme described in Paper I (Del Zanna & Bucciantini 2002) for relativistic hydrodynamics is here extended to the magnetic case by following the strategies prescribed for classical MHD by Londrillo & Del Zanna (2000). The scheme completely avoids spectral decomposition into characteristic waves, which is computationally expensive and subject to many degenerate cases in the magnetic case, while it makes use of a two-speed Riemann solver that just requires the knowledge of the two local fast magnetosonic velocities. Moreover, the onset of spurious magnetic monopoles, which is a typical problem for multi-dimensional MHD upwind codes, is prevented by properly taking into account the solenoidal constraint and the specific antisymmetric nature of the induction equation. Finally, the extension to generalized orthogonal curvilinear coordinate systems is included, thus the scheme is ready to incorporate general relativistic (GRMHD) effects.

Abstract. Multidimensional shock-capturing numerical schemes for special relativistic hydrodynamics (RHD) are computationally more expensive than their correspondent Euler versions, due to the nonlinear relations between conservative and primitive variables and to the consequent complexity of the Jacobian matrices (needed for the spectral decomposition in most of the approximate Riemann solvers of common use). Here an efficient and easy-to-implement three-dimensional (3-D) shockcapturing scheme for ideal RHD is presented. Based on the algorithms developed by P. Londrillo & L. Del Zanna (2000, ApJ, 530, 508) for the non-relativistic magnetohydrodynamic (MHD) case, and having in mind its relativistic MHD extension (to appear in a forthcoming paper), the scheme uses high order (third) Convex Essentially Non-Oscillatory (CENO) finite difference interpolation routines and central-type averaged Riemann solvers, which do not make use of time-consuming characteristic decomposition. The scheme is very efficient and robust, and it gives results comparable to those obtained with more sophisticated algorithms, even in ultrarelativistic multidimensional test problems.

We present a general framework to design Godunov-type schemes for multidimensional ideal magnetohydrodynamic (MHD) systems, having the divergence-free relation and the related properties of the magnetic field B as built-in conditions. Our approach mostly relies on the Constrained Transport (CT) discretization technique for the magnetic field components, originally developed for the linear induction equation, which assures [∇ · B]num = 0 and its preservation in time to within machine accuracy in a finite-volume setting. We show that the CT formalism, when fully exploited, can be used as a general guideline to design the reconstruction procedures of the B vector field, to adapt standard upwind procedures for the momentum and energy equations, avoiding the onset of numerical monopoles of O(1) size, and to formulate approximate Riemann solvers for the induction equation. This general framework will be named here Upwind Constrained Transport (UCT). To demonstrate the versatility of our method, we apply it to a variety of schemes, which are finally validated numerically and compared: a novel implementation for the MHD case of the second order Roe-type positive scheme by Liu and Lax (J. Comp. Fluid Dynam. 5, 133, 1996), and both the second and third order versions of a central-type MHD scheme presented by Londrillo and Del Zanna (Astrophys. J. 530, 508, 2000), where the basic UCT strategies have been first outlined.

Abstract. The structure and the evolution of Pulsar Wind Nebulae (PWNe) are studied by means of two-dimensional axisymmetric relativistic magnetohydrodynamic (RMHD) simulations. After the first imaging of the Crab Nebula with Chandra, a growing number of objects has been found to show in the X-rays spatial features such as rings and jets, that clearly cannot be accounted for within the standard framework of one-dimensional semi-analytical models. The most promising explanation suggested so far is based on the combined effects of the latitude dependence of the pulsar wind energy flux, shaping the wind termination shock and naturally providing a higher equatorial emission, and of the wind magnetization, likely responsible for the jet collimation by hoop stresses downstream of the shock. This scenario is investigated here by following the evolution of a PWN interacting with the confining Supernova Remnant (SNR), from the free expansion to the beginning of the reverberation phase. Our results confirm the oblate shape of the wind termination shock and the formation of a polar jet with supersonic velocities (v ≈ 0.5−0.7c) for high enough values of the equatorial wind magnetization parameter (σ > ∼ 0.01).

A general method for constructing high-order upwind schemes for multidimensional magnetohydrodynamics (MHD), having as a main built-in condition the divergence-free constraint $ AE B \ 0 for the magnetic Ðeld vector B, is proposed. The suggested procedure is based on consistency arguments, by taking into account the speciÐc operator structure of MHD equations with respect to the reference Euler equations of gasdynamics. This approach leads in a natural way to a staggered representation of the B Ðeld numerical data in which the divergence-free condition in the cell-averaged form, corresponding to second-order accurate numerical derivatives, is exactly fulÐlled. To extend this property to higher order schemes, we then give general prescriptions to satisfy a (r ] 1)th order accurate $ AE B \ 0 relation for any numerical B Ðeld having a rth order interpolation accuracy. Consistency arguments lead also to a proper formulation of the upwind procedures needed to integrate the induction equations, assuring the exact conservation in time of the divergence-free condition and the related continuity properties for the B vector components. As an application, a third-order code to simulate multidimensional MHD Ñows of astrophysical interest is developed using essentially nonoscillatoryÈbased reconstruction algorithms. Several test problems to illustrate and validate the proposed approach are Ðnally presented.