Stars Crushed by Black Holes. I. On the Energy Distribution of Stellar Debris in Tidal Disruption Events

The distribution of orbital energies imparted into stellar debris following the close encounter of a star with a supermassive black hole is the principal factor in determining the rate of return of debris to the black hole, and thus in determining the properties of the resulting lightcurves from such events. We present simulations of tidal disruption events for a range of β ≡ r t/r p where r p is the pericenter distance and r t the tidal radius. We perform these simulations at different spatial resolutions to determine the numerical convergence of our models. We compare simulations in which the heating due to shocks is included or excluded from the dynamics. For β ≲ 8, the simulation results are well-converged at sufficiently moderate-to-high spatial resolution, while for β ≳ 8, the breadth of the energy distribution can be grossly exaggerated by insufficient spatial resolution. We find that shock heating plays a non-negligible role only for β ≳ 4, and that typically the effect of shock heating is mild. We show that self-gravity can modify the energy distribution over time after the debris has receded to large distances for all β. Primarily, our results show that across a range of impact parameters, while the shape of the energy distribution varies with β, the width of the energy spread imparted to the bulk of the debris is closely matched to the canonical spread, Δ E = GM • R ⋆ / r t 2 , for the range of β we have simulated.

Numerical convergence of pre-initial conditions on dark matter halo properties

ABSTRACT Generating pre-initial conditions (or particle loads) is the very first step to set up a cosmological N-body simulation. In this work, we revisit the numerical convergence of pre-initial conditions on dark matter halo properties using a set of simulations which only differs in initial particle loads, i.e. grid, glass, and the newly introduced capacity constrained Voronoi tessellation (CCVT). We find that the median halo properties agree fairly well (i.e. within a convergence level of a few per cent) among simulations running from different initial loads. We also notice that for some individual haloes cross-matched among different simulations, the relative difference of their properties sometimes can be several tens of per cent. By looking at the evolution history of these poorly converged haloes, we find that they are usually merging haloes or haloes have experienced recent merger events, and their merging processes in different simulations are out-of-sync, making the convergence of halo properties become poor temporarily. We show that, comparing to the simulation starting with an anisotropic grid load, the simulation with an isotropic CCVT load converges slightly better to the simulation with a glass load, which is also isotropic. Among simulations with different pre-initial conditions, haloes in higher density environments tend to have their properties converged slightly better. Our results confirm that CCVT loads behave as well as the widely used grid and glass loads at small scales, and for the first time we quantify the convergence of two independent isotropic particle loads (i.e. glass and CCVT) on halo properties.

It rains and then? Numerical challenges with the 1D Richards equation in kilometer-resolution land surface modelling

Over the last decade kilometer-scale weather predictions and climate projections have become established. Thereby both the representation of atmospheric processes, as well as land-surface processes need adaptions to the higher-resolution. Soil moisture is a critical variable for determining the exchange of water and energy between the atmosphere and the land surface on hourly to seasonal time scales, and a poor representation of soil processes will eventually feed back on the simulation quality of the atmosphere. Especially the partitioning between infiltration and surface runoff will feed back on the hydrological cycle. Several aspects of the coupled system are affected by a shift to kilometer-scale, convection-permitting models. First of all, the precipitation-intensity distribution changes to more intense events. Second, the time-step of the numerical integration becomes smaller. The aim of this study is to investigate the numerical convergence of the one-dimensional Richards Equation with respect to the soil hydraulic model, vertical layer thickness and time-step during the infiltration process. Both regular and non-regular (unequally spaced) grids typical in land surface modelling are considered, using a conventional semi-implicit vertical discretization. For regular grids, results from a highly idealized experiment on the infiltration process show poor numerical convergence for layer thicknesses larger than approximately 5 cm and for time steps greater than 40 s, irrespective of the soil hydraulic model. The velocity of the wetting front decreases systematically with increasing time step and decreasing vertical resolution. For non-regular grids, a new discretization based on a coordinate transform is introduced. In contrast to simpler vertical discretizations, it is able to represent the solution second-order accurate. The results for non-regular grids are qualitatively similar, as a fast increase in layer thickness with depth is equivalent to a lower vertical resolution. It is argued that the sharp gradients in soil moisture around the propagating wetting front must be resolved properly in order to achieve an acceptable numerical convergence of the Richards Equation. Furthermore, it is shown that the observed poor numerical convergence translates directly into a poor convergence of infiltration-runoff partitioning for precipitation time series characteristic of weather and climate models. As a consequence, soil simulations with low resolution in space and time may produce almost twice the amount of surface runoff within 24 hours than their high-resolution counterparts. Our analysis indicates that the problem is particularly pronounced for kilometer-resolution models.

Numerical Solutions for Systems of Fractional and Classical Integro-Differential Equations via Finite Integration Method Based on Shifted Chebyshev Polynomials

In this paper, the finite integration method and the operational matrix of fractional integration are implemented based on the shifted Chebyshev polynomial. They are utilized to devise two numerical procedures for solving the systems of fractional and classical integro-differential equations. The fractional derivatives are described in the Caputo sense. The devised procedure can be successfully applied to solve the stiff system of ODEs. To demonstrate the efficiency, accuracy and numerical convergence order of these procedures, several experimental examples are given. As a consequence, the numerical computations illustrate that our presented procedures achieve significant improvement in terms of accuracy with less computational cost.

FORMATION OF A MATHEMATICAL MODEL AND ALGORITHM FOR IMPLEMENTATION OF CALCULATION MAIN PIPELINE - OF SHELL STRUCTURE

The paper considers methods of numerical calculation of elements of shell structures - main pipelines under various types of loading. Refined equations of motion for cylindrical shell structures are given. To solve boundary value problems of thin-walled structures, the Bubnov-Galerkin method, finite differences using the sweep method is used. As an example, the solution of the boundary value problem of a cylindrical shell under static loading by the finite difference method is given. And also the results of the study of the numerical convergence of the calculated values.

Validation of variable helix milling instability islands

During machining, it is well-known that unstable self-excited vibrations known as regenerative chatter can limit productivity. There has been a great deal of research that has sought to understand regenerative chatter, and to avoid it through modifications to the machining process. One promising approach is the use of variable helix tools. Here, the time delay between successive tooth passes is intentionally modified, in order to improve the boundary of instability. Previous research has predicted that such tools can offer significant performance improvements whereby islands of instability occur in the stability lobe diagram. By avoiding these islands, it is possible to avoid regenerative chatter, at depths of cut that are orders of magnitude higher than for traditional tools. However, to the authors’ knowledge, these predictions have not been experimentally validated, and there is limited understanding of the parameters that can give rise to these improvements. The present study seeks to address this shortfall. A recent approach to analysing regenerative chatter stability is modified, and its numerical convergence is shown to outperform alternative methods. It is then shown that islands of instability only emerge at relatively high levels of structural damping, and that they are particularly susceptible to model convergence effects. The model predictions are validated against detailed experimental data that uses a specially designed configuration to minimise experimental error. To the authors’ knowledge, this provides the first experimentally validated study of unstable islands in variable helix milling, whilst also demonstrating the importance of structural damping and numerical convergence on the prediction accuracy.