Adaptive Mesh Refinement Calculations Research Articles

The Athena++ Adaptive Mesh Refinement Framework: Design and Magnetohydrodynamic Solvers

Abstract The design and implementation of a new framework for adaptive mesh refinement calculations are described. It is intended primarily for applications in astrophysical fluid dynamics, but its flexible and modular design enables its use for a wide variety of physics. The framework works with both uniform and nonuniform grids in Cartesian and curvilinear coordinate systems. It adopts a dynamic execution model based on a simple design called a “task list” that improves parallel performance by overlapping communication and computation, simplifies the inclusion of a diverse range of physics, and even enables multiphysics models involving different physics in different regions of the calculation. We describe physics modules implemented in this framework for both nonrelativistic and relativistic magnetohydrodynamics (MHD). These modules adopt mature and robust algorithms originally developed for the Athena MHD code and incorporate new extensions: support for curvilinear coordinates, higher-order time integrators, more realistic physics such as a general equation of state, and diffusion terms that can be integrated with super-time-stepping algorithms. The modules show excellent performance and scaling, with well over 80% parallel efficiency on over half a million threads. The source code has been made publicly available.

Level-by-level artificial viscosity and visualization for MHD simulation with adaptive mesh refinement

We propose a numerical method to determine the artificial viscosity in magnetohydrodynamics (MHD) simulations with adaptive mesh refinement (AMR) method, where the artificial viscosity is adaptively changed due to the resolution level of the AMR hierarchy. Although the suitable value of the artificial viscosity depends on the governing equations and the model of target problem, it can be determined by von Neumann stability analysis. By means of the new method, “level-by-level artificial viscosity method,” MHD simulations of Rayleigh–Taylor instability (RTI) are carried out with the AMR method. The validity of the level-by-level artificial viscosity method is confirmed by the comparison of the linear growth rates of RTI between the AMR simulations and the simple simulations with uniform grid and uniform artificial viscosity whose resolution is the same as that in the highest level of the AMR simulation. Moreover, in the nonlinear phase of RTI, the secondary instability is clearly observed where the hierarchical data structure of AMR calculation is visualized as high resolution region floats up like terraced fields. In the applications of the method to general fluid simulations, the growth of small structures can be sufficiently reproduced, while the divergence of numerical solutions can be suppressed.

Synthetic observations of first hydrostatic cores in collapsing low-mass dense cores

First hydrostatic cores are predicted by theories of star formation, but their existence has never been demonstrated convincingly by (sub)millimeter observations. Furthermore, the multiplicity at the early phases of the star formation process is poorly constrained. The purpose of this paper is twofold. First, we seek to provide predictions of ALMA dust continuum emission maps from early Class 0 objects. Second, we show to what extent ALMA will be able to probe the fragmentation scale in these objects. Following our previous paper (Commer\c{c}on et al. 2012, hereafter paper I), we post-process three state-of-the-art radiation-magneto-hydrodynamic 3D adaptive mesh refinement calculations to compute the emanating dust emission maps. We then produce synthetic ALMA observations of the dust thermal continuum from first hydrostatic cores. We present the first synthetic ALMA observations of dust continuum emission from first hydrostatic cores. We analyze the results given by the different bands and configurations and we discuss for which combinations of the two the first hydrostatic cores would most likely be observed. We also show that observing dust continuum emission with ALMA will help in identifying the physical processes occurring within collapsing dense cores. If the magnetic field is playing a role, the emission pattern will show evidence of a pseudo-disk and even of a magnetically driven outflow, which pure hydrodynamical calculations cannot reproduce. The capabilities of ALMA will enable us to make significant progress towards understanding fragmentation at the early Class 0 stage and discovering first hydrostatic cores.

Synthetic observations of first hydrostatic cores in collapsing low-mass dense cores

The low-mass star formation evolutionary sequence is relatively well-defined both from observations and theoretical considerations. The first hydrostatic core is the first protostellar equilibrium object that is formed during the star formation process. Using state-of-the-art radiation-magneto-hydrodynamic 3D adaptive mesh refinement calculations, we aim to provide predictions for the dust continuum emission from first hydrostatic cores. We investigate the collapse and the fragmentation of magnetized one solar mass prestellar dense cores and the formation and evolution of first hydrostatic cores using the RAMSES code. We use three different magnetization levels for the initial conditions, which cover a large variety of early evolutionary morphology, e.g., the formation of a disk or a pseudo-disk, outflow launching, and fragmentation. We post-process the dynamical calculations using the 3D radiative transfer code RADMC-3D. We compute spectral energy distributions and usual evolutionary stage indicators such as bolometric luminosity and temperature. We find that the first hydrostatic core lifetimes depend strongly on the initial magnetization level of the parent dense core. We derive, for the first time, spectral energy distribution evolutionary sequences from high-resolution radiation-magneto-hydrodynamic calculations. We show that under certain conditions, first hydrostatic cores can be identified from dust continuum emission at 24 microns and 70 microns. We also show that single spectral energy distributions cannot help to distinguish between the formation scenarios of the first hydrostatic core, i.e., between the magnetized and non-magnetized models. Spectral energy distributions are a first useful and direct way to target first hydrostatic core candidates but high-resolution interferometry is definitively needed to determine the evolutionary stage of the observed sources.

Adaptive mesh refinement applied to the scalar transported PDF equation in a turbulent jet

AbstractThis paper describes a novel solution method for the transported probability density function (PDF) equation for scalars (compositions). In contrast to conventional solution methods based on the Monte Carlo approach, we use a finite‐volume method combined with adaptive mesh refinement (AMR) applied in both physical and compositional space. The obvious advantage of this over a uniform grid is that fine meshes are only used where the solution requires high resolution. The efficiency of the method is demonstrated by a number of tests involving a turbulent jet flow with up to two scalars (both reacting and non‐reacting). We find that the AMR calculation can be at a fraction of the computer cost of a uniform grid calculation with the same accuracy. Copyright © 2010 John Wiley & Sons, Ltd.

A sub-cell based indicator for troubled zones in RKDG schemes and a novel class of hybrid RKDG+HWENO schemes

Runge–Kutta Discontinuous Galerkin (RKDG) schemes can provide highly accurate solutions for a large class of important scientific problems. Using them for problems with shocks and other discontinuities requires that one has a strategy for detecting the presence of these discontinuities. Strategies that are based on total variation diminishing (TVD) limiters can be problem-independent and scale-free but they can indiscriminately clip extrema, resulting in degraded accuracy. Those based on total variation bounded (TVB) limiters are neither problem-independent nor scale-free. In order to get past these limitations we realize that the solution in RKDG schemes can carry meaningful sub-structure within a zone that may not need to be limited. To make this sub-structure visible, we take a sub-cell approach to detecting zones with discontinuities, known as troubled zones. A monotonicity preserving (MP) strategy is applied to distinguish between meaningful sub-structure and shocks. The strategy does not indiscriminately clip extrema and is, nevertheless, scale-free and problem-independent. It, therefore, overcomes some of the limitations of previously-used strategies for detecting troubled zones. The moments of the troubled zones can then be corrected using a weighted essentially non-oscillatory (WENO) or Hermite WENO (HWENO) approach. In the course of doing this work it was also realized that the most significant variation in the solution is contained in the solution variables and their first moments. Thus the additional moments can be reconstructed using the variables and their first moments, resulting in a very substantial savings in computer memory. We call such schemes hybrid RKDG+HWENO schemes. It is shown that such schemes can attain the same formal accuracy as RKDG schemes, making them attractive, low-storage alternatives to RKDG schemes. Particular attention has been paid to the reconstruction of cross-terms in multi-dimensional problems and explicit, easy to implement formulae have been catalogued for third and fourth order of spatial accuracy. The utility of hybrid RKDG+WENO schemes has been illustrated with several stringent test problems in one and two dimensions. It is shown that their accuracy is usually competitive with the accuracy of RKDG schemes of the same order. Because of their compact stencils and low storage, hybrid RKDG+HWENO schemes could be very useful for large-scale parallel adaptive mesh refinement calculations.

Adaptive Mesh Refinement for conservative systems: multi-dimensional efficiency evaluation

Obtainable computational efficiency is evaluated when using an Adaptive Mesh Refinement (AMR) strategy in time accurate simulations governed by sets of conservation laws. For a variety of 1D, 2D, and 3D hydro- and magnetohydrodynamic simulations, AMR is used in combination with several shock-capturing, conservative discretization schemes. Solution accuracy and execution times are compared with static grid simulations at the corresponding high resolution and time spent on AMR overhead is reported. Our examples reach corresponding efficiencies of 5 to 20 in multi-dimensional calculations and only 1.5–8% overhead is observed. For AMR calculations of multi-dimensional magnetohydrodynamic problems, several strategies for controlling the ∇· B =0 constraint are examined. Three source term approaches suitable for cell-centered B representations are shown to be effective. For 2D and 3D calculations where a transition to a more globally turbulent state takes place, it is advocated to use an approximate Riemann solver based discretization at the highest allowed level(s), in combination with the robust Total Variation Diminishing Lax–Friedrichs method on the coarser levels. This level-dependent use of the spatial discretization acts as a computationally efficient, hybrid scheme.

Highly parallel structured adaptive mesh refinement using parallel language-based approaches

Adaptive Mesh Refinement (AMR) calculations carried out on structured meshes play an exceedingly important role in several areas of science and engineering. This is so not just because AMR techniques allow us to carry out calculations very efficiently but also because they model very precisely the multi-scale fashion in which nature itself works. Many AMR applications are also amongst the most computationally intensive calculations undertaken making it necessary to use parallel supercomputers for their solution. While class library-based approaches are being attempted for parallel AMR we point out here that recent advances in the Fortran 90/95 standard and the OpenMP standard now make it possible to carry out highly parallel AMR calculations using language-based approaches. The language-based approaches offer several advantages over library-based approaches, the two principal ones being portability across parallel platforms and the best possible utilization of Distributed Shared Memory (DSM) hardware on machines that have such hardware. They also free up the applications scientist from being constrained by the static features of a class library. The choice of Fortran also ensures maximal reuse of pre-existing Fortran 77 applications and full Fortran 77-based processing efficiency on each computational node. Our implementation of the ideas presented here in the first author's RIEMANN framework essentially permits any serial, uniform grid, stencil-based Fortran code to be turned into a parallel AMR code. In this paper we first describe our strategy for using Fortran 90 in an object-oriented fashion. This permits AMR applications to be expressed in terms of familiar abstractions that are natural to the process of solving AMR hierarchies. We then describe the OpenMP features that are useful for parallel processing of AMR hierarchies in a load balanced fashion on multiprocessors. The automatic, parallel regridding of AMR hierarchies is also described. We then present a very efficient load balancer and show how it is to be used for load balanced solution of AMR hierarchies. Our load balancer is extremely general and should also see use in other disciplines. We follow this up with the application of the parallel AMR techniques developed here to the solution of elliptic and hyperbolic problems. For our elliptic problem we choose parallel, self-adaptive multigrid as an example. For our hyperbolic problem we choose time-dependent MHD as an example. In either case illustrative information is given about the adaptive processing of these systems. We also provide detailed scalability studies for both the above-mentioned problems which show that our methods scale extremely well up to several hundreds of processors.

The Jeans Condition and Collapsing Molecular Cloud Cores: Filaments or Binaries?

The 1997 and 1998 studies by Truelove and colleagues introduced the Jeans condition as a necessary condition for avoiding artificial fragmentation during protostellar collapse calculations. They found that when the Jeans condition was properly satisfied with their adaptive mesh refinement (AMR) code, an isothermal cloud with an initial Gaussian density profile collapsed to form a thin filament rather than the binary or quadruple protostar systems found in previous calculations. Using a completely different self-gravitational hydrodynamics code introduced by Boss & Myhill in 1992 (B&M), we present here calculations that reproduce the filamentary solution first obtained by Truelove et al. in 1997. The filamentary solution only emerged with very high spatial resolution with the B&M code, with effectively 12,500 radial grid points (R12500). Reproducing the filamentary collapse solution appears to be an excellent means for testing the reliability of self-gravitational hydrodynamics codes, whether grid-based or particle-based. We then show that in the more physically realistic case of an identical initial cloud with nonisothermal heating (calculated in the Eddington approximation with code B&M), thermal retardation of the collapse permits the Gaussian cloud to fragment into a binary protostar system at the same maximum density where the isothermal collapse yields a thin filament. However, the binary clumps soon thereafter evolve into a central clump surrounded by spiral arms containing two more clumps. A roughly similar evolution is obtained using the AMR code with a barotropic equation of state—formation of a transient binary, followed by decay of the binary to form a central object surrounded by spiral arms, though in this case the spiral arms do not form clumps. When the same barotropic equation of state is used with the B&M code, the agreement with the initial phases of the AMR calculation is quite good, showing that these two codes yield mutually consistent results. However, the B&M barotropic result differs significantly from the B&M Eddington result at the same maximum density, demonstrating the importance of detailed radiative transfer effects. Finally, we confirm that even in the case of isothermal collapse, an initially uniform density sphere can collapse and fragment into a binary system, in agreement with the 1998 results of Truelove et al. Fragmentation of molecular cloud cores thus appears to remain as a likely explanation of the formation of binary stars, but the sensitivity of these calculations to the numerical resolution and to the thermodynamical treatment demonstrates the need for considerable caution in computing and interpreting three-dimensional protostellar collapse calculations.