Accurate numerical solutions of the equations of hydrodynamics play an ever more important role in many fields of astrophysics. In this work, we reinvestigate the accuracy of the moving-mesh code Arepo and show how its convergence order can be improved for general problems. In particular, we clarify that for certain problems Arepo only reaches first-order convergence for its original formulation. This can be rectified by simple modifications we propose to the time integration scheme and the spatial gradient estimates of the code, both improving the accuracy of the code. We demonstrate that the new implementation is indeed second-order accurate under the L 1 norm, and in particular substantially improves conservation of angular momentum. Interestingly, whereas these improvements can significantly change the results of smooth test problems, we also find that cosmological simulations of galaxy formation are unaffected, demonstrating that the numerical errors eliminated by the new formulation do not impact these simulations. In contrast, simulations of binary stars followed over a large number of orbital times are strongly affected, as here it is particularly crucial to avoid a long-term build up of errors in angular momentum conservation.

Highly supersonic, compressible turbulence is thought to be of tantamount importance for star formation processes in the interstellar medium. Likewise, cosmic structure formation is expected to give rise to subsonic turbulence in the intergalactic medium, which may substantially modify the thermodynamic structure of gas in virialized dark matter haloes and affect small‐scale mixing processes in the gas. Numerical simulations have played a key role in characterizing the properties of astrophysical turbulence, but thus far systematic code comparisons have been restricted to the supersonic regime, leaving it unclear whether subsonic turbulence is faithfully represented by the numerical techniques commonly employed in astrophysics. Here we focus on comparing the accuracy of smoothed particle hydrodynamics (SPH) and our new moving‐mesh technique AREPO in simulations of driven subsonic turbulence. To make contact with previous results, we also analyse simulations of transsonic and highly supersonic turbulence. We find that the widely employed standard formulation of SPH yields problematic results in the subsonic regime. Instead of building up a Kolmogorov‐like turbulent cascade, large‐scale eddies are quickly damped close to the driving scale and decay into small‐scale velocity noise. Reduced viscosity settings improve the situation, but the shape of the dissipation range differs compared with expectations for a Kolmogorov cascade. In contrast, our moving‐mesh technique does yield power‐law scaling laws for the power spectra of velocity, vorticity and density, consistent with expectations for fully developed isotropic turbulence. We show that large errors in SPH’s gradient estimate and the associated subsonic velocity noise are ultimately responsible for producing inaccurate results in the subsonic regime. In contrast, SPH’s performance is much better for supersonic turbulence, as here the flow is kinetically dominated and characterized by a network of strong shocks, which can be adequately captured with SPH. When compared to fixed grid Eulerian simulations of turbulence, our moving mesh approach shows qualitatively very similar results, although with somewhat better resolving power at the same number of cells, thanks to reduced advection errors and the automatic adaptivity of the AREPO code.

Magnetic fields play an important role in astrophysics on a wide variety of scales, ranging from the Sun and compact objects to galaxies and galaxy clusters. Here we discuss a novel implementation of ideal magnetohydrodynamics (MHD) in the moving mesh code AREPO which combines many of the advantages of Eulerian and Lagrangian methods in a single computational technique. The employed grid is defined as the Voronoi tessellation of a set of mesh-generating points which can move along with the flow, yielding an automatic adaptivity of the mesh and a substantial reduction of advection errors. Our scheme solves the MHD Riemann problem in the rest frame of the Voronoi interfaces using the HLLD Riemann solver. To satisfy the divergence constraint of the magnetic field in multiple dimensions, the Dedner divergence cleaning method is applied. In a set of standard test problems we show that the new code produces accurate results, and that the divergence of the magnetic field is kept sufficiently small to closely preserve the correct physical solution. We also apply the code to two first application problems, namely supersonic MHD turbulence and the spherical collapse of a magnetized cloud. We verify that the code is able to handle both problems well, demonstrating the applicability of this MHD version of AREPO to a wide range of problems in astrophysics.Comment: 11 pages, 9 figures, accepted by MNRA

Solving the Euler equations of ideal hydrodynamics as accurately and efficiently as possible is a key requirement in many astrophysical simulations. It is therefore important to continuously advance the numerical methods implemented in current astrophysical codes, especially also in light of evolving computer technology, which favours certain computational approaches over others. Here we introduce the new adaptive mesh refinement (AMR) code TENET, which employs a high order discontinuous Galerkin (DG) scheme for hydrodynamics. The Euler equations in this method are solved in a weak formulation with a polynomial basis by means of explicit Runge-Kutta time integration and Gauss-Legendre quadrature. This approach offers significant advantages over commonly employed second order finite volume (FV) solvers. In particular, the higher order capability renders it computationally more efficient, in the sense that the same precision can be obtained at significantly less computational cost. Also, the DG scheme inherently conserves angular momentum in regions where no limiting takes place, and it typically produces much smaller numerical diffusion and advection errors than a FV approach. A further advantage lies in a more natural handling of AMR refinement boundaries, where a fall-back to first order can be avoided. Finally, DG requires no wide stencils at high order, and offers an improved data locality and a focus on local computations, which is favourable for current and upcoming highly parallel supercomputers. We describe the formulation and implementation details of our new code, and demonstrate its performance and accuracy with a set of two-and three-dimensional test problems. The results confirm that DG schemes have a high potential for astrophysical applications.

Hydrodynamical simulations of galaxy formation such as the Illustris simulations have progressed to a state where they approximately reproduce the observed stellar mass function from high to low redshift. This in principle allows self-consistent models of reionization that exploit the accurate representation of the diffuse gas distribution together with the realistic growth of galaxies provided by these simulations, within a representative cosmological volume. In this work, we apply and compare two radiative transfer algorithms implemented in a GPU-accelerated code to the 106.5 Mpc wide volume of Illustris in postprocessing in order to investigate the reionization transition predicted by this model. We find that the first generation of galaxies formed by Illustris is just about able to reionize the universe by redshift z ∼ 7, provided quite optimistic assumptions about the escape fraction and the resolution limitations are made. Our most optimistic model finds an optical depth of τ 0.065, which is in very good agreement with recent Planck 2015 determinations. Furthermore, we show that moment-based approaches for radiative transfer with the M1 closure give broadly consistent results with our angular-resolved radiative transfer scheme. In our favoured fiducial model, 20% of the hydrogen is reionized by redshift z = 9.20, and this rapidly climbs to 80% by redshift z = 6.92. It then takes until z = 6.24 before 99% of the hydrogen is ionized. On average, reionization proceeds 'inside-out' in our models, with a size distribution of reionized bubbles that progressively features regions of ever larger size while the abundance of small bubbles stays fairly constant.