Comparison of different nonlinear solvers for 2D time-implicit stellar hydrodynamics

Time-implicit schemes are attractive since they allow numerical time steps that are much larger than those permitted by the Courant-Friedrich-Lewy criterion characterizing time-explicit methods. This advantage comes, however, with a cost: the solution of a system of nonlinear equations is required at each time step. In this work, the nonlinear system results from the discretization of the hydrodynamical equations with the Crank-Nicholson scheme. We compare the cost of different methods, based on Newton-Raphson iterations, to solve this nonlinear system, and benchmark their performances against time-explicit schemes. Since our general scientific objective is to model stellar interiors, we use as test cases two realistic models for the convective envelope of a red giant and a young Sun. Focusing on 2D simulations, we show that the best performances are obtained with the quasi-Newton method proposed by Broyden. Another important concern is the accuracy of implicit calculations. Based on the study of an idealized problem, namely the advection of a single vortex by a uniform flow, we show that there are two aspects: i) the nonlinear solver has to be accurate enough to resolve the truncation error of the numerical discretization, and ii) the time step has be small enough to resolve the advection of eddies. We show that with these two conditions fulfilled, our implicit methods exhibit similar accuracy to time-explicit schemes, which have lower values for the time step and higher computational costs. Finally, we discuss in the conclusion the applicability of these methods to fully implicit 3D calculations.Comment: Accepted for publication in A&

Chaos and Turbulent Nucleosynthesis Prior to a Supernova Explosion

Three-dimensional (3D), time dependent numerical simulations, of flow of matter in stars, now have sufficient resolution to be fully turbulent. The late stages of the evolution of massive stars, leading up to core collapse to a neutron star (or black hole), and often to supernova explosion and nucleosynthesis, are strongly convective because of vigorous neutrino cooling and nuclear heating. Unlike models based on current stellar evolutionary practice, these simulations show a chaotic dynamics characteristic of highly turbulent flow. Theoretical analysis of this flow, both in the Reynolds-averaged Navier-Stokes (RANS) framework and by simple dynamic models, show an encouraging consistency with the numerical results. It may now be possible to develop physically realistic and robust procedures for convection and mixing which (unlike 3D numerical simulation) may be applied throughout the long life times of stars. In addition, a new picture of the presupernova stages is emerging which is more dynamic and interesting (i.e., predictive of new and newly observed phenomena) than our previous one.Comment: 11 pages, 2 figures, Submitted to AIP Advances: Stardust, added figures and modest rewritin

Toward a consistent use of overshooting parametrizations in 1D stellar evolution codes

Several parametrizations for overshooting in 1D stellar evolution calculations coexist in the literature. These parametrizations are used somewhat arbitrarily in stellar evolution codes, based on what works best for a given problem, or even for historical reasons related to the development of each code. We bring attention to the fact that these different parametrizations correspond to different physical regimes of overshooting, depending whether the effects of radiation are dominant, marginal, or negligible. Our analysis is based on previously published theoretical results, as well as multidimensional hydrodynamical simulations of stellar convection where the interaction between the convective region and a stably-stratified region is observed. Although the underlying hydrodynamical processes are the same, the outcome of the overshooting process is profoundly affected by radiative effects. Using a simple picture of the scales involved in the overshooting process, we show how three regimes are obtained, depending on the importance of radiative effects. These three regimes correspond to the different behaviors observed in hydrodynamical simulations so far, and to the three types of parametrizations used in 1D codes. We suggest that the existing parametrizations for overshooting should coexist in 1D stellar evolution codes, and should be applied consistently at convective boundaries depending on the local physical conditions.Comment: 5 pages, 2 figures, to appear in A&A as a regular paper. Last version: language editing + typos in Eq. (6) & (9) correcte

Turbulent convection in stellar interiors. III. Mean-field analysis and stratification effects

We present 3D implicit large eddy simulations (ILES) of the turbulent convection in the envelope of a 5 Msun red giant star and in the oxygen-burning shell of a 23 Msun supernova progenitor. The numerical models are analyzed in the framework of 1D Reynolds-Averaged Navier-Stokes (RANS) equations. The effects of pressure fluctuations are more important in the red giant model, owing to larger stratification of the convective zone. We show how this impacts different terms in the mean-field equations. We clarify the driving sources of kinetic energy, and show that the rate of turbulent dissipation is comparable to the convective luminosity. Although our flows have low Mach number and are nearly adiabatic, our analysis is general and can be applied to photospheric convection as well. The robustness of our analysis of turbulent convection is supported by the insensitivity of the mean-field balances to linear mesh resolution. We find robust results for the turbulent convection zone and the stable layers in the oxygen-burning shell model, and robust results everywhere in the red giant model, but the mean fields are not well converged in the narrow boundary regions (which contain steep gradients) in the oxygen-burning shell model. This last result illustrates the importance of unresolved physics at the convective boundary, which governs the mixing there.Comment: 26 pages, 20 figures, Accepted for publication in Ap

The First 3D Simulations of Carbon Burning in a Massive Star

We present the first detailed three-dimensional hydrodynamic implicit large eddy simulations of turbulent convection for carbon burning. The simulations start with an initial radial profile mapped from a carbon burning shell within a 15 solar mass stellar evolution model. We considered 4 resolutions from 128^3 to 1024^3 zones. These simulations confirm that convective boundary mixing (CBM) occurs via turbulent entrainment as in the case of oxygen burning. The expansion of the boundary into the surrounding stable region and the entrainment rate are smaller at the bottom boundary because it is stiffer than the upper boundary. The results of this and similar studies call for improved CBM prescriptions in 1D stellar evolution models.Comment: 5 pages, 3 figures. Published in IAUS 329 on 28/07/1

Linking 1D Evolutionary to 3D Hydrodynamical Simulations of Massive Stars

Stellar evolution models of massive stars are important for many areas of astrophysics, for example nucleosynthesis yields, supernova progenitor models and understanding physics under extreme conditions. Turbulence occurs in stars primarily due to nuclear burning at different mass coordinates within the star. The understanding and correct treatment of turbulence and turbulent mixing at convective boundaries in stellar models has been studied for decades but still lacks a definitive solution. This paper presents initial results of a study on convective boundary mixing (CBM) in massive stars. The 'stiffness' of a convective boundary can be quantified using the bulk Richardson number ($\textrm{Ri}_B$), the ratio of the potential energy for restoration of the boundary to the kinetic energy of turbulent eddies. A 'stiff' boundary ($\textrm{Ri}_B \sim 10^4$) will suppress CBM, whereas in the opposite case a 'soft' boundary ($\textrm{Ri}_B \sim 10$) will be more susceptible to CBM. One of the key results obtained so far is that lower convective boundaries (closer to the centre) of nuclear burning shells are 'stiffer' than the corresponding upper boundaries, implying limited CBM at lower shell boundaries. This is in agreement with 3D hydrodynamic simulations carried out by Meakin and Arnett [The Astrophysical Journal 667:448-475, 2007]. This result also has implications for new CBM prescriptions in massive stars as well as for nuclear burning flame front propagation in Super-Asymptotic Giant Branch stars and also the onset of novae.Comment: Accepted for publication (12/12/15) in the Physica Scripta focus issue on Turbulent Mixing and Beyon

On the relevance of bubbles and potential flows for stellar convection

Recently Pasetto et al. have proposed a new method to derive a convection theory appropriate for the implementation in stellar evolution codes. Their approach is based on the simple physical picture of spherical bubbles moving within a potential flow in dynamically unstable regions, and a detailed computation of the bubble dynamics. Based on this approach the authors derive a new theory of convection which is claimed to be parameter free, non-local and time-dependent. This is a very strong claim, as such a theory is the holy grail of stellar physics. Unfortunately we have identified several distinct problems in the derivation which ultimately render their theory inapplicable to any physical regime. In addition we show that the framework of spherical bubbles in potential flows is unable to capture the essence of stellar convection, even when equations are derived correctly.Comment: 14 pages, 3 figures. Accepted for publication in Monthly Notices of the Royal Astronomical Society. (Comments and criticism are welcomed

Linking 1D Stellar Evolution to 3D Hydrodynamical Simulations

In this contribution we present initial results of a study on convective boundary mixing (CBM) in massive stellar models using the GENEVA stellar evolution code. Before undertaking costly 3D hydrodynamic simulations, it is important to study the general properties of convective boundaries, such as the: composition jump; pressure gradient; and `stiffness'. Models for a 15Mo star were computed. We found that for convective shells above the core, the lower (in radius or mass) boundaries are `stiffer' according to the bulk Richardson number than the relative upper (Schwarzschild) boundaries. Thus, we expect reduced CBM at the lower boundaries in comparison to the upper. This has implications on flame front propagation and the onset of novae.Comment: 2 pages, 1 figure. To appear in proceedings of the IAU Symposium 307: New Windows on Massive Stars: Asteroseismology, Interferometry and Spectropolarimetr