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Tidal disruption events involve numerous physical processes (fluid dynamics, magnetohydrodynamics, radiation transport, self-gravity, general relativistic dynamics) in highly nonlinear ways, and, because TDEs are transients by definition, frequently in non-equilibrium states. For these reasons, numerical solution of the relevant equations can be an essential tool for studying these events. In this chapter, we present a summary of the key problems of the field for which simulations offer the greatest promise and identify the capabilities required to make progress on them. We then discuss what has been—and what cannot be—done with existing numerical methods. We close with an overview of what methods now under development may do to expand our ability to understand these events.

Accretion disks orbiting black holes power high-energy systems such as X-ray binaries and Active Galactic Nuclei. Observations are providing increasingly detailed quantitative information about such systems. This data has been interpreted using standard toy-models that rely on simplifying assumptions such as regular flow geometry and a parameterized stress. Global numerical simulations offer a way to investigate the basic physical dynamics of accretion flows without these assumptions and, in principle, lead to a genuinely predictive theory. In recent years we have developed a fully three-dimensional general relativistic magnetohydrodynamic simulation code that evolves time-dependent inflows into Kerr black holes. Although the resulting global simulations of black hole accretion are still somewhat simplified, they have brought to light a number of interesting results. These include the formation of electro-magnetically dominated jets powered by the black hole's rotation, and the presence of strong stresses in the plunging region of the accretion flow. The observational consequences of these features are gradually being examined. Increasing computer power and increasingly sophisticated algorithms promise a bright future for the computational approach to black hole accretion.

Supermassive black holes (SMBHs) in the centers of massive galaxies are thought to predominantly grow in brief Eddington-rate quasar phases accompanied by starbursts, but on-going starbursts in luminous quasars are difficult to observe. Buried under the natural coronagraph, obscured quasars offer a unique window for direct, robust host-galaxy spectroscopy otherwise virtually inaccessible for luminous quasars. Our pilot study at z ~ 0.5 (Liu et al. 2009) revealed a substantial contribution from very young stellar populations with ages less than ~ 100 Myr in all of the observed host galaxy spectra. More dramatically, in three out of the nine SDSS quasars observed, we have witnessed strong infant starbursts with ages of ~ 5 Myr, clocked by the telltale Wolf–Rayet emission features.

Quasi-Periodic Erupters (QPEs) are a remarkable class of objects exhibiting very large amplitude quasi-periodic X-ray flares. Although numerous dynamical models have been proposed to explain them, relatively little attention has been given to using the properties of their radiation to constrain their dynamics. Here we show that the observed luminosity, spectrum, repetition period, duty cycle, and fluctuations in the latter two quantities point toward a model in which: a main sequence star on a moderately eccentric orbit around a supermassive black hole periodically transfers mass to the Roche lobe of the black hole; orbital dynamics lead to mildly-relativistic shocks near the black hole; and thermal X-rays at the observed temperature are emitted by the gas as it flows away from the shock. Strong X-ray irradiation of the star by the flare itself augments the mass transfer, creates fluctuations in flare timing, and stirs turbulence in the stellar atmosphere that amplifies magnetic field to a level at which magnetic stresses can accelerate infall of the transferred mass toward the black hole.

The tidal disruption event AT2019dsg was observed from radio to X-rays and was possibly accompanied by a high-energy neutrino. Previous interpretations have focused on continued injection by a central engine as the source of energy for radio emission. We show that continuous energy injection is unnecessary; the radio data can be explained by a single ejection of plasma that supplies all the energy needed. To support this assertion, we analyze the synchrotron self-absorbed spectra in terms of the equipartition model. Similar to previous analyses, we find that the energy in the radio-emitting region increases approximately \\(\\propto t^{0.7}\\) and the lengthscale of this region grows \\(\\propto t\\) at a rate \\(\\simeq0.06c\\). This event resembles the earliest stage of a supernova remnant: because the ejected mass is much greater than the shocked external mass, its velocity remains unchanged, while the energy in shocked gas grows with time. The radio-emitting material gains energy from the outflow, not continuing energy injection by the central object. Although energy injection from an accreting BH cannot be completely excluded, the energy injection rate is very different from the fallback luminosity, and maintaining constant outflow velocity requires fine-tuning demanding further physical explanation. If the neutrino association is real, the energy injection needed is much greater than for the radio emission, suggesting that the detected neutrino did not arise from the radio-emitting region.

New studies indicate that if the emission lines of active galactic nuclei are powered by the continuum radiation, fluctuations in the continuum drive correspond with fluctuations in the local line emissivity within the emission-line region.

The magnetorotational instability (MRI) has been extensively studied in circular magnetized disks, and its ability to drive accretion has been demonstrated in a multitude of scenarios. There are reasons to expect eccentric magnetized disks to also exist, but the behavior of the MRI in these disks remains largely uncharted territory. Here we present the first simulations that follow the nonlinear development of the MRI in eccentric disks. We find that the MRI in eccentric disks resembles circular disks in two ways, in the overall level of saturation and in the dependence of the detailed saturated state on magnetic topology. However, in contrast with circular disks, the Maxwell stress in eccentric disks can be negative in some disk sectors, even though the integrated stress is always positive. The angular momentum flux raises the eccentricity of the inner parts of the disk and diminishes the same of the outer parts. Because material accreting onto a black hole from an eccentric orbit possesses more energy than material tracing the innermost stable circular orbit, the radiative efficiency of eccentric disks may be significantly lower than circular disks. This may resolve the \"inverse energy problem\" seen in many tidal disruption events.

Black hole - neutron star \\((BH/NS)\\) binaries are of interest in many ways: they are intrinsically multi-messenger systems, highly transient, radiate gravitational waves detectable by LIGO, and may produce \\(\\gamma\\)-ray bursts. Although it has long been assumed that their late-stage orbital evolution is driven entirely by gravitational wave emission, we show here that in certain circumstances, mass transfer from the neutron star onto the black hole can both alter the binary's orbital evolution and significantly reduce the neutron star's mass when the fraction of its mass transferred per orbit is \\(\\gtrsim 10^{-2}\\), the neutron star's mass diminishes by order-unity, leading to mergers in which the neutron star mass is exceptionally small. The mass transfer creates a gas disk around the black hole \\({\\it before}\\) merger that can be comparable in mass to the debris remaining after merger, i.e. \\(\\sim 0.1 M_\\odot\\). These processes are most important when the initial neutron star/black hole mass ratio \\(q\\) is in the range \\(\\approx 0.2 - 0.8\\), the orbital semimajor axis is \\(40 \\lesssim a_0/r_g \\lesssim 300 \\) (\\(r_g \\equiv GM_{\\rm B}/c^2\\)), and the eccentricity is large, \\(e_0 \\gtrsim 0.8\\). Systems of this sort may be generated through the dynamical evolution of a triple system, as well as by other means.

Tidal disruption events involve numerous physical processes (fluid dynamics, magnetohydrodynamics, radiation transport, self-gravity, general relativistic dynamics) in highly nonlinear ways, and, because TDEs are transients by definition, frequently in non-equilibrium states. For these reasons, numerical solution of the relevant equations can be an essential tool for studying these events. In this chapter, we present a summary of the key problems of the field for which simulations offer the greatest promise and identify the capabilities required to make progress on them. We then discuss what has been---and what cannot be---done with existing numerical methods. We close with an overview of what methods now under development may do to expand our ability to understand these events.

We present a series of simulations in both pure hydrodynamics (HD) and magnetohydrodynamics (MHD) exploring the degree to which alignment of disks subjected to external precessional torques (e.g., as in the `Bardeen-Petterson' effect) is dependent upon the disk sound speed c_s. Across the range of sound speeds examined, we find that the influence of the sound speed can be encapsulated in a simple \"lumped-parameter\" model proposed by Sorathia et al. (2013a). In this model, alignment fronts propagate outward at a speed ~0.2 rOmega_precess(r), where Omega_precess is the local test-particle precession frequency. Meanwhile, transonic radial motions transport angular momentum both inward and outward at a rate that may be described roughly in terms of an orientation diffusion model with diffusion coefficient ~2c_s^2/Omega, for local orbital frequency Omega. The competition between the two leads, in isothermal disks, to a stationary position for the alignment front at a radius proportional to c_s^(-4/5). For alignment to happen at all, the disk must either be turbulent due to the magnetorotational instability in MHD, or, in HD, it must be cool enough for the bending waves driven by disk warp to be nonlinear at their launch point. Contrary to long-standing predictions, warp propagation in MHD disks is diffusive independent of the parameter c_s/(alpha v_orb$, for orbital speed v_orb and ratio of stress to pressure of alpha. In purely HD disks, i.e., those with no internal stresses other than bulk viscosity, warmer disks align weakly or not at all; cooler disks align qualitatively similarly to MHD disks.