Implicit large eddy simulations of anisotropic weakly compressible turbulence with application to core-collapse supernovae
- David Radice^{1}Email author,
- Sean M Couch^{1} and
- Christian D Ott^{1}
https://doi.org/10.1186/s40668-015-0011-0
© Radice et al. 2015
Received: 15 April 2015
Accepted: 12 August 2015
Published: 21 August 2015
Abstract
In the implicit large eddy simulation (ILES) paradigm, the dissipative nature of high-resolution shock-capturing schemes is exploited to provide an implicit model of turbulence. The ILES approach has been applied to different contexts, with varying degrees of success. It is the de-facto standard in many astrophysical simulations and in particular in studies of core-collapse supernovae (CCSN). Recent 3D simulations suggest that turbulence might play a crucial role in core-collapse supernova explosions, however the fidelity with which turbulence is simulated in these studies is unclear. Especially considering that the accuracy of ILES for the regime of interest in CCSN, weakly compressible and strongly anisotropic, has not been systematically assessed before. Anisotropy, in particular, could impact the dissipative properties of the flow and enhance the turbulent pressure in the radial direction, favouring the explosion. In this paper we assess the accuracy of ILES using numerical methods most commonly employed in computational astrophysics by means of a number of local simulations of driven, weakly compressible, anisotropic turbulence. Our simulations employ several different methods and span a wide range of resolutions. We report a detailed analysis of the way in which the turbulent cascade is influenced by the numerics. Our results suggest that anisotropy and compressibility in CCSN turbulence have little effect on the turbulent kinetic energy spectrum and a Kolmogorov \(k^{-5/3}\) scaling is obtained in the inertial range. We find that, on the one hand, the kinetic energy dissipation rate at large scales is correctly captured even at low resolutions, suggesting that very high “effective Reynolds number” can be achieved at the largest scales of the simulation. On the other hand, the dynamics at intermediate scales appears to be completely dominated by the so-called bottleneck effect, i.e., the pile up of kinetic energy close to the dissipation range due to the partial suppression of the energy cascade by numerical viscosity. An inertial range is not recovered until the point where high resolution ∼512^{3}, which would be difficult to realize in global simulations, is reached. We discuss the consequences for CCSN simulations.
Keywords
1 Introduction
Despite decades of studies and compelling evidence that a significant fraction (Clausen et al. 2015) of stars with initial masses in excess of ∼8 solar masses explode as core-collapse supernovae (CCSN) at the end of their evolution, the exact details of the explosion mechanism are still uncertain (Woosley and Janka 2005; Janka et al. 2012; Burrows 2013; Foglizzo et al. 2015). Current state-of-the art 3D simulations either fail to explode or have explosion energies that fall short of the observed energies by factors of a few for most of the progenitor mass range (Janka 2012; Burrows 2013; Foglizzo et al. 2015).
The dynamics at the center of a star undergoing core collapse is shaped by a delicate balance between competing effects where all of the known forces: gravity, electromagnetism, weak and strong interactions, are important. The task of modeling these systems is made particularly challenging by the fact that the generation of the asymptotic explosion energies, although enormous (\({\sim}10^{44}~\mathrm{J}\)), requires a rather subtle, percent-level imbalance between non-linear processes over many dynamical times.
The flow of plasma in the core of a star going supernova is known to be unstable to convection (Herant 1995; Burrows et al. 1995; Janka and Müller 1996; Foglizzo et al. 2006) and/or to another large scale instability known as standing accretion shock instability (Blondin et al. 2003; Foglizzo et al. 2007). In any case, given the very large Reynolds numbers, as large as ∼10^{17} in the region of interest (Abdikamalov et al. 2015) (the so-called gain region, where neutrino heating dominates over neutrino cooling), it is expected that the resulting flow will be fully turbulent. It has been suggested (Murphy et al. 2013; Couch and Ott 2015) recently that turbulence and, in particular, turbulent pressure could tip the balance of the forces in favor of explosion. In this respect, anisotropy is of key importance, because it results in an effective radial pressure support with adiabatic index \(\gamma_{\mathrm{turb}} = 2\), much larger than that of thermal (radiation) pressure (\(\gamma_{\mathrm{th}} \simeq4/3\)). This means that turbulent kinetic energy is a much more valuable source of radial pressure support than thermal energy (see Appendix).
All of the current numerical simulations employ the implicit large eddy simulation (ILES) paradigm (Garnier et al. 2000; Grinstein et al. 2011) (also known as monotone integrated LES (MILES)) of exploiting the dissipative nature of high resolution shock capturing (HRSC) methods as an implicit turbulence model. However, the combination of the use of rather dissipative schemes and the relatively low spatial resolution that can be achieved in global simulations is such that the fidelity with which turbulence is captured is questionable (Abdikamalov et al. 2015).
To be useful in the context of CCSN simulations, an ILES should, at the very least, account for the right rate of decay of the kinetic energy at the largest scales while avoiding unphysical pile up of energy at smaller scales. Unfortunately, all of the current simulations seem to be strongly dominated by the so-called bottleneck effect (Abdikamalov et al. 2015), which corresponds to an inefficient energy transfer across intermediate scales due to the viscous suppression of non-linear interaction with smaller scales (Yakhot and Zakharov 1993; She and Jackson 1993; Falkovich 1994; Verma and Donzis 2007; Frisch et al. 2008). Current global simulations achieve resolutions, in the turbulent region, comparable to those of 30^{3}-70^{3} lattices in periodic domains (Couch and O’Connor 2014; Couch and Ott 2015; Abdikamalov et al. 2015). At these resolutions, almost all of the dynamical range of the simulations can be expected to be directly affected by numerical viscosity (Sytine et al. 2000). The fidelity with which turbulence is captured in these simulations will then depend on the degree with which the numerical truncation error approximates an LES closure.
In this respect, it has been shown by Garnier et al. (1999) and Johnsen et al. (2010) that many HRSC methods can be too dissipative to yield a faithful description of turbulence at low resolutions. These studies, however, considered a different regime, decaying isotropic turbulence, while turbulence in a core-collapse supernova, as well as in many other astrophysical settings, is often strongly anisotropic (Arnett et al. 2009; Murphy et al. 2013; Couch and Ott 2015) as rotational invariance is broken by gravity. Garnier et al. (1999) and Johnsen et al. (2010) also considered different numerical schemes with respect to those used in supernova simulations. Both of these aspects can, in principle, be important. First of all, strong anisotropies could potentially influence the turbulence dynamics at the level of the energy cascade and of the dissipation (Casciola et al. 2007). Secondly, some of the schemes used in computational astrophysics, such as the piecewise parabolic method (PPM) (Colella and Woodward 1984) as well as some of the MUSCL (Toro 1999) schemes, have been shown, differently from some of the methods considered by Garnier et al. (1999) and Johnsen et al. (2010), to be well suited for ILES (Schmidt et al. 2006; Thornber et al. 2007).
The aim of this work is to fill the gap between existing theoretical studies and the particular applications of our interest. To this end we use a publicly available code, FLASH (Fryxell et al. 2000; Dubey et al. 2009; Lee et al. 2014), which is widely used in the computational astrophysics community, and perform a series of simulations of turbulence in a regime relevant for core-collapse supernovae: driven at large scale, with large anisotropies and mildly compressible. We use five different numerical setups and, for each, several resolutions in the range from 64^{3} to 512^{3} in a periodic domain. We study in detail the way in which the energy cascade across different scales is represented by our ILES and we discuss the use of local or lower dimensional diagnostics that can be used to assess the quality of a global simulation in a complex geometry where 3D spectra are not readily available.
The rest of this paper is organized as follows. First, in Section 2, we discuss the exact setup of our simulations and the diagnostic quantities used in our analysis. Then, in Section 3, we discuss the basic characteristics of the flow realized in our simulations. In Section 4, we present a detailed analysis of the way in which the energy cascade is captured by the different schemes at different scales. In particular, we quantify the accuracy with which different methods capture the decay rate of energy from the largest scales and the way in which energy is distributed across scales. We discuss the role of anisotropies in the context of the \(4/5\)-law, a fundamental exact relation for isotropic and incompressible turbulence relating the statistics of velocity fluctuations with the energy dissipation rate (see Section 2.3), in Section 5. We explore the use of the 2D, transverse, energy spectrum as a diagnostic for 3D simulations in Section 6. Finally, we present a brief summary of our main findings, as well as a discussion of their implications for CCSN simulations in Section 7. Appendix contains some supplemental background material on the role of turbulence in the explosion mechanism of CCSN.
2 Methods
2.1 Numerical methods
Equations (1) and (2) are solved using the directionally-unsplit hydrodynamics solver of the open-source FLASH simulation framework. FLASH implements the corner transport upwind method (Colella 1990) for fully directionally-unsplit evolution of the Euler equations (Lee and Deane 2009; Lee 2013). FLASH includes several options for the order of spatial reconstruction (Lee et al. 2014), including 2nd-order TVD (Toro 1999), 3rd-order PPM (Colella and Woodward 1984), and 5th-order WENOZ (Borges et al. 2008). Fluxes are computed at 2nd-order accuracy using one of a number of approximate Riemann solvers included in FLASH, such as HLLE (Einfeldt 1988) and HLLC (Toro et al. 1994). Second-order accuracy in time is achieved via a characteristic tracing evolution of the Riemann solver input states to the time step midpoint (Colella and Woodward 1984). We remark that, in accordance with the ILES, paradigm, we do not include any additional sub-grid scale model, but relied on the implicit turbulent closure built in the numerical schemes we use for the integration of the hydrodynamics equation.
2.2 Energy transfer equations
2.3 Structure functions
The energy spectrum and its sources/fluxes give a comprehensive picture of the energy cascade and can be used to assess the level of convergence of the simulation. Unfortunately, 3D energy spectra and fluxes are not easily accessible in calculations in complex domains and/or with inhomogeneous turbulence. In these cases, local quantities in the physical domain are more easily extracted and analyzed. Hence, one of the goals of this work is to validate the use of indirect measures of convergence of ILES. Among these quantities, the structure functions of the velocity appear to be natural candidates for study.
2.4 Transverse energy spectrum
As was the case for the 3D spectra, also here the spectrum is non-trivial only for integer \(k_{y}\) and \(k_{z}\), when periodicity is taken into account. The integral in equation (26) is treated analogously to the integral in the equation (14) for the 3D case, while the average in the x-direction in equation (27) is converted to an average over the x-extent of the simulation box.
3 Basic flow properties
We employ the finite-volume HRSC (Godunov) approach in which physical states are reconstructed at inter-cell boundaries and local Riemann problems are solved to compute the physical inter cell fluxes. In particular, we perform five groups of simulations using different numerical methods. Each group is labeled using the name of the reconstruction algorithm and of the Riemann solver. For instance TVD_HLLE, denotes a group of simulations done using TVD reconstruction and HLLE Riemann solver. Single simulations are labeled using their resolution so that, for instance, TVD_HLLE_N128, denotes the TVD_HLLE run done using a 128^{3} grid. For all of the runs the timestep is chosen to have a CFL, i.e., \(c \Delta t / \Delta x\), of 0.4, c being the maximum characteristics speed, with the exception of the PPM_HLLC_CFL0.8 runs where we set the CFL to 0.8. For the TVD runs we use the monotonized central (MC) slope limiter (Toro 1999). The runs with PPM use the original flattening and artificial viscosity prescriptions from Colella and Woodward (1984). The artificial viscosity coefficient is 0.1. We remark that the use of the artificial viscosity for PPM is not really necessary in this regime (Porter and Woodward 1994), however our goal is not to perform a study of the turbulent dynamics, but to assess how each numerical method performs when used under the same condition as in a real CCSN simulation where strong shocks need to be handled in some parts of the domain.
4 The energy cascade
In this section we focus our analysis on the accuracy with which the energy cascade is captured by our ILES runs. First, we focus on the largest scales of the simulation with the goal of quantifying the accuracy in the decay rate of the energy as a function of the resolution for the different methods. Next, we will look at the energy distribution at smaller scales where, in resolved simulations, the inertial range starts. Finally, we will look at the dynamics in the dissipation region and summarize.
4.1 Energy decay rate
In the context of CCSN simulations this means that the large scale kinetic energy, a crucial quantity for the dynamics of the explosion (Couch and Ott 2015), can be faithfully captured even with simulations achieving modest Reynolds numbers.
As discussed before, we expect that \(\Pi(k) \simeq\langle\epsilon \rangle\) over an extended region in Fourier space should be a direct indication that a simulation has been able to recover an inertial range. Perhaps not surprisingly, in light of previous results (Sytine et al. 2000), we find that regions where \(\Pi\simeq\langle\epsilon\rangle\) as wide as a few wave numbers \(4 \lesssim k \lesssim10\) only appear at the highest resolutions (we will discuss the inertial range in more detail in Section 4.2). However, the amount of energy decaying from the largest scales reaches an asymptotic value much quicker than that implying that the total kinetic energy budget at the largest scales is well resolved even at modest resolutions.
4.2 Energy spectra
Looking at any of the groups of runs in Figure 5, one can immediately notice that the spectra obtained at different resolutions do not collapse into a single curve in the dissipation region, as would be required by Kolmogorov’s first similarity hypothesis (Frisch 1996) (cf. Gotoh et al. (2002)). This lack of convergence in the dissipation region could be due to the non-linear viscosity of HRSC schemes. This, in turn, could result in an anomalous scaling of η with the grid spacing. Such scaling has been reported in the past for ILES, but it is not very well understood (Aspden et al. 2009). The good agreement between the three different groups of simulations employing the HLLC Riemann solver seems to support this hypothesis and suggests that the nonlinear viscosity introduced by the Riemann solver is an important ingredient in setting this scaling.
Convergence appears to be recovered at larger scales \(\gtrsim8 \Delta x\) (\(512 k \Delta x \lesssim64\)), but the spectra appear to be dominated by the bottleneck effect. This manifests itself as a bump in the compensated spectra extending from the dissipation range until the end of the inertial range, for the simulations that show one (e.g., until \(512 k \Delta x = 10\) for the HLLC runs), or until the energy injection scale (\(512 k \Delta x = 4\)), for the simulations that show no or little inertial range (TVD_HLLE). The bottleneck effect is a viscous phenomenon which is also observed in direct numerical simulations. However, in the present context where viscosity is of numerical origin, it is at the very least questionable if a pronounced bottleneck is a desirable feature of the modeling. In astrophysical flows, where the Reynolds numbers are typically very large, this pile up of energy at large scales is unphysical and could affect the quantitative and qualitative outcome of a simulation (Abdikamalov et al. 2015). A quantification of the bottleneck effect in terms of the energy budget is discussed in Section 4.4.
At even larger scales, an inertial range (\(E\sim k^{-5/3}\) and \(\Pi \sim \mathrm{const}\), see Figure 3) seems to be recovered by the least dissipative schemes (PPM and WENOZ with HLLC) in the region \(4\lesssim k \lesssim10\). PPM_HLLE and TVD_HLLE have a more limited region, a few wave numbers at most, that could be interpreted as being an inertial range. We note that this resolution is not particularly high in comparison with state of the art DNS (Kaneda et al. 2003; Federrath 2013), but it would already correspond to an extremely high resolution in global CCSN simulations that typically have ∼30-70 zones across the turbulent region (Abdikamalov et al. 2015).
The overall behavior of the spectra, as obtained by all schemes, is consistent with Kolmogorov’s theory of turbulence. The anisotropic contributions to the angle-integrated spectra are too small to be detected in our data.
4.3 Numerical viscosity
The first thing to notice is that the numerical viscosity provided by all numerical schemes is not constant, but differs by roughly an order of magnitude between low and high k. Having a wave number dependent viscosity is a desirable feature expected in any LES model (explicit or otherwise). Nevertheless, this makes the definition and calculation of the effective Reynolds number achieved in a simulation ambiguous. Meaningful ways to estimate it for ILES have been proposed (Zhou et al. 2014) and they can be used to ease the comparison between different simulations and assess their quality. However, one has to be very careful while using any quoted “Reynolds number” from an ILES, to estimate things like the dynamical range achieved by a simulation, because the dissipative properties of ILES differ considerably from the ones of the true Navier-Stokes equations.
Two other features can be observed in most of the numerical viscosity profiles. First, many of them exhibit a sudden reversal at high wave numbers. This is due to the fact that the numerical viscosity does not behave like a shear viscosity so that, although the numerical diffusion is strong at those scales, the numerical viscosity appears small because of a partial decoupling between vorticity and dissipation. Second, at high resolution and at the largest scales, the numerical viscosity is close to zero or even slightly negative. The reason is that the residual of equation (9) oscillates around zero and it is too small to be reliably extracted from our data: a much longer integration time would be needed to accumulate enough statistics for it.
Finally, a comparison between the numerical viscosity reveals two interesting effects. First, by comparing PPM_HLLC and PPM_ HLLE, we see that the choice of the Riemann solver affects the viscosity at basically all scales. Second, if we compare PPM_HLLC, PPM_HLLC_ CFL0.8 and WENOZ_HLLC, we see that doubling the timestep appears to have an effect comparable to the difference between the PPM and WENOZ reconstructions at intermediate scales (\(40\lesssim k \lesssim100\)).
4.4 The energy distribution
So far we have been concerned with the energy decay rate from the largest scales, which we have shown to be well captured by the ILES (Section 4.1), and with the energy transfer in the inertial range, which we have seen to be described accurately only at much higher resolutions (Section 4.2). In a turbulent flow both of these aspects are important and a good ILES should display a distribution of energy across vortical structures at different scales that is as close as possible to the asymptotic one. Obviously, there is a limit to the accuracy that any ILES can achieve at a fixed resolution. Here, we make this statement more quantitative by considering the amount of kinetic energy that is well resolved by each simulation at a given resolution.
5 The \(4/5\)-law
The \(4/5\)-law (equation (25)) is not a-priori valid in the regime of turbulence we are considering. However, the \(4/5\)-law has been numerically verified to hold also in some situations outside the domain of validity of its derivation. For instance, for isotropic mildly compressible decaying (Porter et al. 2002) and driven (Benzi et al. 2008) turbulence. In the anisotropic case, however, anisotropic contributions cannot be excluded (Biferale et al. 2002), although they are known to be subdominant in some important cases (Calzavarini et al. 2002; Biferale et al. 2003; Kaneda et al. 2008). In this section we show that equation (25) is consistent with our data over a wide range of scales.
6 The transverse spectrum
Finally, we want to comment on the use of 2D transverse spectra in 3D simulations, a practice typically employed in the analysis of turbulence in CCSN simulations (Dolence et al. 2013; Couch and O’Connor 2014; Handy et al. 2014; Abdikamalov et al. 2015).
Abdikamalov et al. (2015) concluded, also based on the analysis of 2D spectra, that turbulence in CCSN simulations is dominated by the bottleneck effect. Given the resolutions used in CCSN studies, our work supports their conclusion. However, in the light of Figure 9, we recommend that future studies supplement the analysis of 2D spectra with 3rd-order structure functions, that, as we have shown, can give a more accurate description of the energy cascade.
7 Conclusions
The details of the explosion mechanism of CCSNe have eluded our comprehension in spite of more than 50 years of studies (Woosley and Janka 2005; Janka et al. 2012; Burrows 2013; Foglizzo et al. 2015). Recent numerical advances (Murphy et al. 2013; Couch and Ott 2013; Couch and Ott 2015; Müller and Janka 2015) suggest that turbulence might play a fundamental role in tipping over the balance of the forces and lead to successful explosions (see also Appendix). At the same time, the level of accuracy of current simulations, which employ the ILES methodology, is unclear (Abdikamalov et al. 2015). Turbulence in CCSNe is mildly compressible, but strongly anisotropic (Murphy et al. 2013; Couch and Ott 2015). Simulations use rather dissipative numerical schemes (because they have to deal with strong shock waves and complex microphysics) and relatively low resolution, a combination (anisotropic turbulence and dissipative schemes) that has not been systematically studied before.
With the goal of assessing the reliability of ILES employed in the study of CCSNe, as well as in other areas of physics and astrophysics, we performed a series of local simulations of driven, anisotropic, weakly compressible turbulence. We compared five commonly employed numerical schemes with different reconstruction methods, Riemann solvers, and time step size. Each was run at 4 different resolutions ranging from 64^{3} to 512^{3}. Our analysis focused on the fidelity with which the turbulent cascade is represented in each model. In particular, we performed an analysis both in Fourier space (with the velocity power-spectra and the energy flux) and in physical space (with the 3rd-order structure functions). Finally, we measured the numerical viscosity of each scheme from the residual of the specific kinetic energy equation.
We found that, on the one hand, all of the numerical setups are able to accurately capture the decay rate of kinetic energy from the injection scale, with errors at the few % level already at 128^{3} (e.g., \({\sim}2.5\%\) for PPM_HLLC_N128). On the other hand, a large fraction of the energy is at unresolved scales where it piles up due to the bottleneck effect and an inertial range appears only at the highest resolutions (512^{3}). Even at this resolution, which would be difficult to achieve in global simulations, only roughly \({\sim}80\%\) (the exact number depends on the scheme, see Section 4.4) of the energy is resolved, the remaining \({\sim}20\%\) accumulates as excess energy at intermediate scales (the cumulative energy excess at the grid scale alone is as large as \({\sim}5\%\) of the total energy).
Current CCSN simulations have resolutions of at most of 30-70 points covering the gain region (Abdikamalov et al. 2015) (the energy injection scale). Based on our analysis we expect that at these resolutions even the energy decay rate from the largest scales will not be completely converged, but will show errors of up to tens of percent, depending on the numerical scheme (see Section 4.1). At smaller scales, the dynamics is going to be completely dominated by the bottleneck effect. This is in agreement with the findings of Abdikamalov et al. (2015), based on the use of global simulations reaching a maximum resolution of 66 grid points covering radially the extent of the gain region.
Based on our findings, we expect that, if the resolution in global simulations is increased by a factor ∼2 from the one of Abdikamalov et al. (2015), the decay rate will be converged to within a few % of the asymptotic value. This implies that the ratio between thermal and kinetic energy, a crucial quantity for the onset of the explosion, will also be converged to within a few %, at least when the energy injection rate changes slowly compared to the eddy turnover time (which is roughly \({\sim}20 \mathrm{~ms}\) in a CCSN (Ott et al. 2013; Couch and O’Connor 2014)). Unfortunately, while the lead up to explosion occurs over a larger timescale of a few hundred milliseconds, the transition to explosion can happen over much shorter timescales (one turnover time or less) (Couch and Ott 2015). This means that the dynamics of the cascade over smaller time and length scales in the gain region also needs to be captured correctly since changes in the energy input rate on such short time scales will yield an inaccurate representation of the energy on large scales due to the bottleneck effect. This could require an increase of resolution by a factor ∼4-8 with respect to current high-resolution simulations. Additional work using semi-global or global simulations will be required to more firmly establish the resolutions requirements at the transition of the explosion.
Concerning the properties of anisotropic turbulence in our simulations, we found anisotropy contributions to the energy spectrum and to the angle-averaged formulation of the \(4/5\)-law to be subdominant: the accuracy with which the \(4/5\)-law is satisfied is limited only by the employed resolution and the energy spectrum appears to be consistent with Kolmogorov \(k^{-5/3}\) scaling. We also found the transverse energy spectrum with respect to the direction of anisotropy, a quantity typically computed in CCSN simulations, to overestimate the bottleneck with respect to the angle-integrated 3D spectrum. For this reason, we recommend future studies of CCSN to supplement (or replace) the analysis of the transverse spectrum with the analysis of the 3rd order, angle-integrated, structure function (or, where possible, with the 3D spectrum itself).
Our results are, of course, dependent on the choice of the numerical scheme. In particular, we found significant differences in the dissipative properties of schemes employing the HLLC Riemann solver with respect to schemes using the more dissipative HLLE solver. The reconstruction order of the scheme is also important, although, while significant differences are found between TVD and PPM, the differences between PPM and WENOZ are much more minute (despite WENOZ being significantly more computationally expensive than PPM). In the end, none of the schemes we considered seems to be able to yield an accurate representation of the kinetic energy distribution across different scales at an affordable resolution for global CCSN simulations. A possible way forward would be to adopt low-dissipation numerical schemes especially designed for the use in ILES, such as the methods proposed by Hickel et al. (2006); Martín et al. (2006) or Thornber et al. (2008). Implementing and testing these schemes will be subject of future work.
An important limitation of the present work is that we considered a very idealized setup. On the one hand, this allows us to benchmark the behavior of ILES in a controlled environment. On the other hand, our simulations cannot fully capture all features of the turbulent convective flow in a CCSN. Unlike the situation in a CCSN, our local simulations did not include a vertical advective velocity field that is due to the accretion of the stellar mantle. However, the advective velocities are nearly constant in the regions of interest and Galilean invariance ensures that our results are unaffected. More limiting is the local nature of our simulations and the inevitable choice of boundary conditions. Moreover, our simulations could not take into account spatial variations in gravity and the large-scale radial convergence of the flow in globally spherical problems like collapsing stars. Addressing these issues will also be subject of future work.
Declarations
Acknowledgements
We acknowledge helpful discussions with E Abdikamalov, WD Arnett, A Burrows, R Fisher, C Meakin, P Mösta, J Murphy, M Norman, and L Roberts. This research was partially supported by the National Science Foundation under award nos. AST-1212170 and PHY-1151197 and by the Sherman Fairchild Foundation. The simulations were performed on the Caltech compute cluster Zwicky (NSF MRI-R2 award no. PHY-0960291), on the NSF XSEDE network under allocation TG-PHY100033, and on NSF/NCSA BlueWaters under NSF PRAC award no. ACI-1440083. The software used in this work was in part developed by the DOE NNSA-ASC OASCR Flash Center at the University of Chicago.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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