A numerical study of the Einstein field equations, based on the 3 + 1 foliation of the spacetime, is presented. A pseudo-spectral technique has been employed for simulations in vacuum, within two different formalisms, namely the Arnowitt–Deser–Misner (ADM) and the conformal Baumgarte–Shapiro–Shibata–Nakamura (BSSN) approach. The numerical code is based on the Fourier decomposition, accompanied by different filtering techniques. The role of the dealiasing, as well as the influence of the filter type, has been investigated. The algorithms have been stabilized via a novel procedure that controls self-consistently the regularity of the solutions. The accuracy of the model has been validated through standard testbeds, revealing that the filtered pseudo-spectral technique is among the most accurate approaches. Finally, the procedure has been stressed via black hole dynamics and a new strategy, based on hyperviscous dissipation that suppresses spurious boundary problems, has been proposed. The model represents a valid tool of investigation, particularly suitable for the inspection of small scale nonlinear phenomena in gravitational dynamics.

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The microphysical, kinetic properties of astrophysical plasmas near accreting compact objects are still poorly understood. For instance, in modern general-relativistic magnetohydrodynamic simulations, the relation between the temperature of electrons T e and protons T p is prescribed in terms of simplified phenomenological models where the electron temperature is related to the proton temperature in terms of the ratio between the gas and magnetic pressures, or the β parameter. We here present a very comprehensive campaign of two-dimensional kinetic particle-in-cell simulations of special-relativistic turbulence to investigate systematically the microphysical properties of the plasma in the transrelativistic regime. Using a realistic mass ratio between electrons and protons, we analyze how the index of the electron energy distributions κ, the efficiency of nonthermal particle production  , and the temperature ratio  := T e / T p vary over a wide range of values of β and σ. For each of these quantities, we provide two-dimensional fitting functions that describe their behavior in the relevant space of parameters, thus connecting the microphysical properties of the plasma, κ,  , and  , with the macrophysical ones β and σ. In this way, our results can find application in a wide range of astrophysical scenarios, including the accretion and the jet emission onto supermassive black holes, such as M87* and Sgr A*.

The microphysical, kinetic properties of astrophysical plasmas near accreting compact objects are still poorly understood. For instance, in modern general-relativistic magnetohydrodynamic simulations, the relation between the temperature of electrons T e and protons T p is prescribed in terms of simplified phenomenological models where the electron temperature is related to the proton temperature in terms of the ratio between the gas and magnetic pressures, or β parameter. We here present a very comprehensive campaign of two-dimensional kinetic Particle-In-Cell (PIC) simulations of special-relativistic turbulence to investigate systematically the microphysical properties of the plasma in the trans-relativistic regime. Using a realistic mass ratio between electrons and protons, we analyze how the index of the electron energy distributions κ, the efficiency of nonthermal particle production E, and the temperature ratio T := T e /T p , vary over a wide range of values of β and σ. For each of these quantities, we provide two-dimensional fitting functions that describe their behaviour in the relevant space of parameters, thus connecting the microphysical properties of the plasma, κ, E, and T , with the macrophysical ones β and σ. In this way, our results can find application in wide range of astrophysical scenarios, including the accretion and the jet emission onto supermassive black holes, such as M87* and Sgr A*.