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High-order Godunov methods for gas dynamics have become a standard tool for simulating different classes of astrophysical flows. Their accuracy is mostly determined by the spatial interpolant used to reconstruct the pair of Riemann states at cell interfaces and by the Riemann solver that computes the interface fluxes. In most Godunov-type methods, these two steps can be treated independently, so that many different schemes can in principle be built from the same numerical framework. Because astrophysical simulations often test out the limits of what is feasible with the computational resources available, it is essential to find the scheme that produces the numerical solution with the desired accuracy at the lowest computational cost. However, establishing the best combination of numerical options in a Godunov-type method to be used for simulating a complex hydrodynamic problem is a nontrivial task. In fact, formally more accurate schemes do not always outperform simpler and more diffusive methods, especially if sharp gradients are present in the flow. For this work, we used our fully compressible Seven-League Hydro SLH ) code to test the accuracy of six reconstruction methods and three approximate Riemann solvers on two- and three-dimensional (2D and 3D) problems involving subsonic flows only. We considered Mach numbers in the range from $10^ $ to $10^ $, which are characteristic of many stellar and geophysical flows. In particular, we considered a well-posed, 2D, Kelvin--Helmholtz instability problem and a 3D turbulent convection zone that excites internal gravity waves in an overlying stable layer. Although the different combinations of numerical methods converge to the same solution with increasing grid resolution for most of the quantities analyzed here, we find that (i) there is a spread of almost four orders of magnitude in computational cost per fixed accuracy between the methods tested in this study, with the most performant method being a combination of a low-dissipation Riemann solver and a sextic reconstruction scheme; (ii) the low-dissipation solver always outperforms conventional Riemann solvers on a fixed grid when the reconstruction scheme is kept the same; (iii) in simulations of turbulent flows, increasing the order of spatial reconstruction reduces the characteristic dissipation length scale achieved on a given grid even if the overall scheme is only second order accurate; (iv) reconstruction methods based on slope-limiting techniques tend to generate artificial, high-frequency acoustic waves during the evolution of the flow; and (v) unlimited reconstruction methods introduce oscillations in the thermal stratification near the convective boundary, where the entropy gradient is steep.
High-order Godunov methods for gas dynamics have become a standard tool for simulating different classes of astrophysical flows. Their accuracy is mostly determined by the spatial interpolant used to reconstruct the pair of Riemann states at cell interfaces and by the Riemann solver that computes the interface fluxes. In most Godunov-type methods, these two steps can be treated independently, so that many different schemes can in principle be built from the same numerical framework. Because astrophysical simulations often test out the limits of what is feasible with the computational resources available, it is essential to find the scheme that produces the numerical solution with the desired accuracy at the lowest computational cost. However, establishing the best combination of numerical options in a Godunov-type method to be used for simulating a complex hydrodynamic problem is a nontrivial task. In fact, formally more accurate schemes do not always outperform simpler and more diffusive methods, especially if sharp gradients are present in the flow. For this work, we used our fully compressible Seven-League Hydro SLH ) code to test the accuracy of six reconstruction methods and three approximate Riemann solvers on two- and three-dimensional (2D and 3D) problems involving subsonic flows only. We considered Mach numbers in the range from $10^ $ to $10^ $, which are characteristic of many stellar and geophysical flows. In particular, we considered a well-posed, 2D, Kelvin--Helmholtz instability problem and a 3D turbulent convection zone that excites internal gravity waves in an overlying stable layer. Although the different combinations of numerical methods converge to the same solution with increasing grid resolution for most of the quantities analyzed here, we find that (i) there is a spread of almost four orders of magnitude in computational cost per fixed accuracy between the methods tested in this study, with the most performant method being a combination of a low-dissipation Riemann solver and a sextic reconstruction scheme; (ii) the low-dissipation solver always outperforms conventional Riemann solvers on a fixed grid when the reconstruction scheme is kept the same; (iii) in simulations of turbulent flows, increasing the order of spatial reconstruction reduces the characteristic dissipation length scale achieved on a given grid even if the overall scheme is only second order accurate; (iv) reconstruction methods based on slope-limiting techniques tend to generate artificial, high-frequency acoustic waves during the evolution of the flow; and (v) unlimited reconstruction methods introduce oscillations in the thermal stratification near the convective boundary, where the entropy gradient is steep.
We conduct one-dimensional stellar evolutionary numerical simulations under the assumption that an efficient dynamo operates in the core of massive stars years to months before core collapse and find that the magnetic activity enhances mass-loss rate and might trigger binary interaction that leads to outbursts. We assume that the magnetic flux tubes that the dynamo forms in the inner core buoy out to the outer core where there is a steep entropy rise and a molecular weight drop. There the magnetic fields turn to thermal energy, i.e. by reconnection. We simulate this energy deposition where the entropy steeply rises and find that for our simulated cases the envelope radius increases by a factor of ≃1.2–2 and luminosity by about an order of magnitude. These changes enhance the mass-loss rate. The envelope expansion can trigger a binary interaction that powers an outburst. Because magnetic field amplification depends positively on the core rotation rate and operates in cycles, not in all cases the magnetic activity will be powerful enough to change envelope properties. Namely, only a fraction of core-collapse supernovae experiences pre-explosion outbursts.
The treatment of convection remains a major weakness in the modelling of stellar evolution with one-dimensional (1D) codes. The ever increasing computing power makes now possible to simulate in 3D part of a star for a fraction of its life, allowing us to study the full complexity of convective zones with hydrodynamics codes. Here, we performed state-of-the-art hydrodynamics simulations of turbulence in a neon-burning convective zone, during the late stage of the life of a massive star. We produced a set of simulations varying the resolution of the computing domain (from 1283 to 10243 cells) and the efficiency of the nuclear reactions (by boosting the energy generation rate from nominal to a factor of 1000). We analysed our results by the mean of Fourier transform of the velocity field, and mean-field decomposition of the various transport equations. Our results are in line with previous studies, showing that the behaviour of the bulk of the convective zone is already well captured at a relatively low resolution (2563), while the details of the convective boundaries require higher resolutions. The different boosting factors used show how various quantities (velocity, buoyancy, abundances, abundance variances) depend on the energy generation rate. We found that for low boosting factors, convective zones are well mixed, validating the approach usually used in 1D stellar evolution codes. However, when nuclear burning and turbulent transport occur on the same timescale, a more sophisticated treatment would be needed. This is typically the case when shell mergers occur.
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