Full length articleNumerical approaches for multidimensional simulations of stellar explosions
Introduction
Multidimensional simulations shed light on how fluid instabilities arising in supernovae (SNe) mix ejecta (Herant and Woosley, 1994, Joggerst et al., 2009, Joggerst et al., 2010, Joggerst and Whalen, 2011). Unfortunately, computing fully self-consistent 3D stellar evolution models, from their formation to collapse, for the explosion setup is still beyond the realm of contemporary computational power. One alternative is to first evolve the main sequence star in a 1D stellar evolution code in which the equations of momentum, energy and mass are solved on a spherically-symmetric grid, such as KEPLER (Weaver et al., 1978) or MESA (Paxton et al., 2011). Once the star reaches the pre-supernova phase, its 1D profiles can then be mapped into multidimensional hydro codes such as CASTRO (Almgren et al., 2010, Zhang et al., 2011) or FLASH (Fryxell et al., 2000) and continue to be evolved until the star explodes.
Differences between codes in dimensionality and coordinate mesh can lead to numerical issues such as violation of conservation of mass and energy when profiles are mapped from one code to another. A first, simple approach could be to initialize multidimensional grids by linear interpolation from corresponding mesh points on the 1D profiles. However, linear interpolation becomes invalid when the new grid fails to resolve critical features in the original profile such as the inner core of a star. This is especially true when porting profiles from 1D Lagrangian codes, which can easily resolve very small spatial features in mass coordinate, to a fixed or adaptive Eulerian grid. In addition to conservation laws, some physical processes such as nuclear burning are very sensitive to temperature, so errors in mapping can lead to very different outcomes for the simulations such as altering the nucleosynthesis and energetics of SNe. None address the conservation of physical quantities by such procedures. We examine these issues and introduce a new scheme for mapping 1D data sets to multidimensional grids.
Seeding the pre-supernova profile of the star with realistic perturbations is important to illuminate how fluid instabilities later erupt and mix the star during the explosion. Massive stars usually develop convective zones prior to exploding as SNe (Woosley et al., 2002, Heger and Woosley, 2002). Multidimensional stellar evolution models suggest that the fluid inside the convective regions can be highly turbulent (Porter and Woodward, 2000, Arnett and Meakin, 2011). However, in lieu of the 3D stellar evolution calculations necessary to produce such perturbations from first principles, multidimensional simulations are usually just seeded with random perturbations. In reality, if the star is convective and the fluid in those zones is turbulent (Davidson, 2004), a better approach is to imprint the multidimensional profiles with velocity perturbations with a Kolmogorov energy spectrum (Frisch, 1995).
In addition to implementing realistic initial conditions, care must be taken to determine the resolution that multidimensional simulations require to resolve the most important physical scales and yield consistent results given the computational resources that are available. We provide a systematic approach for finding this resolution for multidimensional stellar explosions. The structure of the paper is as follows; in Section 2 we describe the key features of the KEPLER and CASTRO codes. We describe our initial mapping scheme and demonstrate it by porting a massive star model from KEPLER to CASTRO in Section 3. We review our scheme for seeding 2D and 3D stellar profiles with turbulent perturbations and present hydrodynamic simulations done with these profiles in CASTRO in Section 4. We provide a strategy for finding the proper resolution for multidimensional simulations with multiscale processes such as hydrodynamics and nuclear burning in Section 5 and conclude the results in Section 6.
Section snippets
Stellar model
We model the evolution of main sequence stars with KEPLER (Weaver et al., 1978), a 1D Lagrangian stellar evolution code. KEPLER solves the evolution equations for mass, momentum, and energy, including relevant physical processes such as nuclear burning and mixing due to convection. When the star reaches the pre-supernova phase (hundreds of seconds prior to launching the SN shock), we map its 1D profiles onto a multidimensional grid in CASTRO. When the star explodes, its initial spherical
Conservative mapping
Since the star is very nearly in hydrostatic equilibrium and we want to conserve total energy, care must be taken when mapping its profile from the uniform Lagrangian grid in mass coordinate to the new Eulerian spatial grid. Zingale et al. (2002) and Mocák et al. (2009) have also studied mapping 1D initial conditions onto multidimensional grids. Different from our scheme, they focus on maintaining the hydrostatic equilibrium setup, because hydrostatic equilibrium is required for their
Initial perturbation
Seeding the pre-supernova profile of the star with realistic perturbations may be important to understanding how fluid instabilities later erupt and mix the star during the explosion. Massive stars usually develop convective zones prior to exploding as SNe (Woosley et al., 2002, Heger and Woosley, 2002). Multidimensional stellar evolution models suggest that the fluid inside the convective regions can be highly turbulent (Porter and Woodward, 2000, Arnett and Meakin, 2011). However, in lieu of
Resolving the early stages of the explosion
In addition to implementing realistic initial conditions and relevant physics for CASTRO, care must be taken to determine the resolution of multidimensional simulations required to resolve the most important physical scales and yield consistent results, given the computational resources that are available. We provide a systematic approach for finding this resolution for multidimensional stellar explosions.
Simulations that include nuclear burning, which governs nucleosynthesis and the energetics
Conclusion
Multidimensional stellar evolution and supernova simulations are numerically challenging because multiple physical processes (hydrodynamics, gravity, burning) occur on many scales in space and time. For computational efficiency, 1D stellar models are often used as initial conditions in 2D and 3D calculations. Mapping 1D profiles onto multidimensional grids can introduce serious numerical artifacts, one of the most severe of which is the violation of conservation of physical quantities. We have
Acknowledgments
The authors thank anonymous referees for reviewing this manuscript and providing many insightful comments, the members of the CCSE at LBNL for help with CASTRO, and Hank Childs for assistance with VISIT. We also thank Volker Bromm, Dan Kasen, Lars Bildsten, John Bell, Adam Burrows, and Stan Woosley for many useful discussions. K.C. was supported by the IAU-Gruber Fellowship, Stanwood Johnston Fellowship, and KITP Graduate Fellowship. A.H. was supported by a future fellowship from the Australian
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