Philipp Edelmann

Heidelberg Institute for Theoretical Studies, Germany

- spherical symmetry
- no dynamical effects
- turbulence model with free parameters

- no enforced symmetry
- full equations of fluid dynamics
- turbulence from first principles

Mach number $M = \frac{u}{c} = \frac{\text{fluid velocity}}{\text{speed of sound}}$

Flows in the stellar interior are usually at low Mach numbers.

speed of sound $c = \sqrt{\gamma \frac{p}{\rho}} \propto \sqrt{\frac{T}{\mu}}$

Mach | $10^{-1}$ | $10^{-2}$ | $10^{-3}$ |

prec. Roe | |||

Roe |

- modify underlying equations
- e.g. anelastic approximation, Maestro, …

works well for flows with only low Mach numbers

intermediate Mach numbers ($\sim 10^{-1}$) or mixed case needs the full Euler equations

- Cartesian grids are badly adapted to spherical stars
- Spherical grids have singularities (center, axis)
- Map Cartesian computational grid to curvilinear grid
- Code stays simple, geometry encoded in
*metric*terms

explicit | implicit |
---|---|

time step constraint for stability $\Delta t_\text{explicit} \le \CFL{} \frac{\Delta x}{|u + c|} \stackrel{u \ll c}{\approx} \CFL{} \frac{\Delta x}{c}$ sound crosses one cell per step |
time step constraint for accuracy $\Delta t_\text{implicit} \le \CFL{} \frac{\Delta x}{|u|}$ fluid crosses one cell per step |

- Implicit time steps are larger by a factor of $1/M$.
- At each step a non-linear system has to be solved using Newton–Raphson.
- We need iterative linear solvers to invert the huge Jacobian.
- In SLH implicit time-stepping is more efficient for $M\lesssim0.1$.

collaborators:

Raphael Hirschi (Keele) and Cyril Georgy (Geneva)

Friedrich Röpke (HITS), Leonhard Horst (Würzburg)

Maeder (2009)

image credit: Maeder (2009), originally Talon (1997)

- should become shear unstable in stellar evolution code
- should not show other instabilities at the same time
- ideally similar time scale in stellar evolution and hydro code

- $20\,M_\odot$ ZAMS star, 40% crit. rotation
- core O burning phase
- Ne burning shell
- convectively stable
- Ri unstable

- 2D equatorial plane
- more than 6 hours of physical time
- special mapping of GENEC data to keep convective stability

by Leonhard Horst

- not straightforward to map model to 3D
- strict shellular rotation cannot always be upheld, while keeping a stable model
- some modifications to $\Omega$ profile to get a stable model in the equatorial plane

$\bar{A}$

collaborators:

Alexander Heger (Monash), Friedrich Röpke (HITS)

- zero metallicity initial model
- core He burning produced already significant amount of $^{12}\text{C}$
- H burning shell with $X(^{12}\text{C})=10^{-9}$
- convective core grows, reaches H burning shell

$250\,\msol$ star ($Z=0$) during core He burning

$128^3$ grid for about 4 days

Mach number

^{14}N

energy release

$512^3$ grid

- start simulation right before overshooting reaches H shell
- include core using cubed sphere

- In many aspects stars should be treated as 3D, dynamical objects.
- SE codes are still needed to cover evolutionary timescale.
- Low Mach numbers require special numerical methods.
- Fully implicit, 3D hydro is possible and scales well to large supercomputers.
- We can now look at many poorly understood phenomena from stellar evolution in greater detail using hydro simulations.