A. Nonaka

CASTRO: A NEW COMPRESSIBLE ASTROPHYSICAL SOLVER. I. HYDRODYNAMICS AND SELF-GRAVITY

We present a new code, CASTRO, that solves the multicomponent compressible hydrodynamic equations for astrophysical flows including self-gravity, nuclear reactions, and radiation. CASTRO uses an Eulerian grid and incorporates adaptive mesh refinement (AMR). Our approach to AMR uses a nested hierarchy of logically rectangular grids with simultaneous refinement in both space and time. The radiation component of CASTRO will be described in detail in the next paper, Part II, of this series.

MAESTRO: AN ADAPTIVE LOW MACH NUMBER HYDRODYNAMICS ALGORITHM FOR STELLAR FLOWS

Many astrophysical phenomena are highly subsonic, requiring specialized numerical methods suitable for long-time integration. In a series of earlier papers we described the development of MAESTRO, a low Mach number stellar hydrodynamics code that can be used to simulate long-time, low-speed flows that would be prohibitively expensive to model using traditional compressible codes. MAESTRO is based on an equation set derived using low Mach number asymptotics; this equation set does not explicitly track acoustic waves and thus allows a significant increase in the time step. MAESTRO is suitable for two- and three-dimensional local atmospheric flows as well as three-dimensional full-star flows. Here, we continue the development of MAESTRO by incorporating adaptive mesh refinement (AMR). The primary difference between MAESTRO and other structured grid AMR approaches for incompressible and low Mach number flows is the presence of the time-dependent base state, whose evolution is coupled to the evolution of the full solution. We also describe how to incorporate the expansion of the base state for full-star flows, which involves a novel mapping technique between the one-dimensional base state and the Cartesian grid, as well as a number of overall improvements to the algorithm. We examine the efficiency and accuracy of our adaptive code, and demonstrate that it is suitable for further study of our initial scientific application, the convective phase of Type Ia supernovae.

HIGH-RESOLUTION SIMULATIONS OF CONVECTION PRECEDING IGNITION IN TYPE Ia SUPERNOVAE USING ADAPTIVE MESH REFINEMENT

We extend our previous three-dimensional, full-star simulations of the final hours of convection preceding ignition in Type Ia supernovae to higher resolution using the adaptive mesh refinement capability of our low Mach number code, MAESTRO. We report the statistics of the ignition of the first flame at an effective 4.34 km resolution and general flow field properties at an effective 2.17 km resolution. We find that off-center ignition is likely, with radius of 50 km most favored and a likely range of 40-75 km. This is consistent with our previous coarser (8.68 km resolution) simulations, implying that we have achieved sufficient resolution in our determination of likely ignition radii. The dynamics of the last few hot spots preceding ignition suggest that a multiple ignition scenario is not likely. With improved resolution, we can more clearly see the general flow pattern in the convective region, characterized by a strong outward plume with a lower speed recirculation. We show that the convective core is turbulent with a Kolmogorov spectrum and has a lower turbulent intensity and larger integral length scale than previously thought (on the order of 16 km s–1 and 200 km, respectively), and we discuss the potential consequences for the first flames.

MULTIDIMENSIONAL MODELING OF TYPE I X-RAY BURSTS. I. TWO-DIMENSIONAL CONVECTION PRIOR TO THE OUTBURST OF A PURE 4He ACCRETOR

We present multidimensional simulations of the early convective phase preceding ignition in a Type I X-ray burst using the low Mach number hydrodynamics code, MAESTRO. A low Mach number approach is necessary in order to perform long-time integration required to study such phenomena. Using MAESTRO, we are able to capture the expansion of the atmosphere due to large-scale heating while capturing local compressibility effects such as those due to reactions and thermal diffusion. We also discuss the preparation of one-dimensional initial models and the subsequent mapping into our multidimensional framework. Our method of initial model generation differs from that used in previous multidimensional studies, which evolved a system through multiple bursts in one dimension before mapping onto a multidimensional grid. In our multidimensional simulations, we find that the resolution necessary to properly resolve the burning layer is an order of magnitude greater than that used in the earlier studies mentioned above. We characterize the convective patterns that form and discuss their resulting influence on the state of the convective region, which is important in modeling the outburst itself.

THE CONVECTIVE PHASE PRECEDING TYPE Ia SUPERNOVAE

The convective flow in the moments preceding the explosion of a Type Ia supernova determines where the initial flames that subsequently burn through the star first ignite. We continue our exploration of the final hours of this convection using the low Mach number hydrodynamics code, MAESTRO. We present calculations exploring the effects of slow rotation and show diagnostics that examine the distribution of likely ignition points. In the current calculations, we see a well-defined convection region persist up to the point of ignition, and we see that even a little rotation is enough to break the coherence of the convective flow seen in the radial velocity field. Our results suggest that off-center ignition may be favored, with ignition ranging out to a radius of 100 km and a maximum likelihood of ignition at a radius around 50 km.

Efficient variable-coefficient finite-volume stokes solvers

We investigate several robust preconditioners for solving the saddle-point linear systems that arise from spatial discretization of unsteady and steady variable-coefficient Stokes equations on a uniform staggered grid. Building on the success of using the classical projection method as a preconditioner for the coupled velocity-pressure system [B. E. Griffith, J. Comp. Phys., 228 (2009), pp. 75657595], as well as established techniques for steady and unsteady Stokes flow in the finite-element literature, we construct preconditioners that employ independent generalized Helmholtz and Poisson solvers for the velocity and pressure subproblems. We demonstrate that only a single cycle of a standard geometric multigrid algorithm serves as an effective inexact solver for each of these subproblems. Contrary to traditional wisdom, we find that the Stokes problem can be solved nearly as efficiently as the independent pressure and velocity subproblems, making the overall cost of solving the Stokes system comparable to the cost of classical projection or fractional step methods for incompressible flow, even for steady flow and in the presence of large density and viscosity contrasts. Two of the five preconditioners considered here are found to be robust to GMRES restarts and to increasing problem size, making them suitable for large-scale problems. Our work opens many possibilities for constructing novel unsplit temporal integrators for finite-volume spatial discretizations of the equations of low Mach and incompressible flow dynamics.

Low Mach number fluctuating hydrodynamics of diffusively mixing fluids

We formulate low Mach number fluctuating hydrodynamic equations appropriate for modeling diffusive mixing in isothermal mixtures of fluids with different density and transport coefficients. These equations eliminate the fluctuations in pressure associated with the propagation of sound waves by replacing the equation of state with a local thermodynamic constraint. We demonstrate that the low Mach number model preserves the spatio-temporal spectrum of the slower diffusive fluctuations. We develop a strictly conservative finite-volume spatial discretization of the low Mach number fluctuating equations in both two and three dimensions and construct several explicit Runge-Kutta temporal integrators that strictly maintain the equation of state constraint. The resulting spatio-temporal discretization is second-order accurate deterministically and maintains fluctuation-dissipation balance in the linearized stochastic equations. We apply our algorithms to model the development of giant concentration fluctuations in the presence of concentration gradients, and investigate the validity of common simplifications such as neglecting the spatial non-homogeneity of density and transport properties. We perform simulations of diffusive mixing of two fluids of different densities in two dimensions and compare the results of low Mach number continuum simulations to hard-disk molecular dynamics simulations. Excellent agreement is observed between the particle and continuum simulations of giant fluctuations during time-dependent diffusive mixing.

LOW MACH NUMBER MODELING OF CONVECTION IN HELIUM SHELLS ON SUB-CHANDRASEKHAR WHITE DWARFS. I. METHODOLOGY

We assess the robustness of a low Mach number hydrodynamics algorithm for modeling helium shell convection on the surface of a white dwarf in the context of the sub-Chandrasekhar model for Type Ia supernovae. We use the low Mach number stellar hydrodynamics code, MAESTRO, to perform three-dimensional, spatially-adaptive simulations of convection leading up to the point of the ignition of a burning front. We show that the low Mach number hydrodynamics model provides a robust description of the system.

Low Mach Number Fluctuating Hydrodynamics of Multispecies Liquid Mixtures

We develop a low Mach number formulation of the hydrodynamic equations describing transport of mass and momentum in a multispecies mixture of incompressible miscible liquids at specified temperature and pressure, which generalizes our prior work on ideal mixtures of ideal gases [Balakrishnan et al., “Fluctuating hydrodynamics of multispecies nonreactive mixtures,” Phys. Rev. E 89 013017 (2014)] and binary liquid mixtures [Donev et al., “Low mach number fluctuating hydrodynamics of diffusively mixing fluids,” Commun. Appl. Math. Comput. Sci. 9(1), 47-105 (2014)]. In this formulation, we combine and extend a number of existing descriptions of multispecies transport available in the literature. The formulation applies to non-ideal mixtures of arbitrary number of species, without the need to single out a “solvent” species, and includes contributions to the diffusive mass flux due to gradients of composition, temperature, and pressure. Momentum transport and advective mass transport are handled using a low Mach number approach that eliminates fast sound waves (pressure fluctuations) from the full compressible system of equations and leads to a quasi-incompressible formulation. Thermal fluctuations are included in our fluctuating hydrodynamics description following the principles of nonequilibrium thermodynamics. We extend the semi-implicit staggered-grid finite-volume numerical method developed in our prior work on binary liquid mixtures [Nonaka et al., “Low mach number fluctuating hydrodynamics of binary liquid mixtures,” arXiv:1410.2300 (2015)] and use it to study the development of giant nonequilibrium concentration fluctuations in a ternary mixture subjected to a steady concentration gradient. We also numerically study the development of diffusion-driven gravitational instabilities in a ternary mixture and compare our numerical results to recent experimental measurements [Carballido-Landeira et al., “Mixed-mode instability of a miscible interface due to coupling between Rayleigh–Taylor and double-diffusive convective modes,” Phys. Fluids 25, 024107 (2013)] in a Hele-Shaw cell. We find that giant nonequilibrium fluctuations can trigger the instability but are eventually dominated by the deterministic growth of the unstable mode, in both quasi-two-dimensional (Hele-Shaw) and fully three-dimensional geometries used in typical shadowgraph experiments.

LOW MACH NUMBER MODELING OF CORE CONVECTION IN MASSIVE STARS

This work presents three-dimensional simulations of core convection in a 15 M ☉ star halfway through its main sequence lifetime. To perform the necessary long-time calculations, we use the low Mach number code MAESTRO, with initial conditions taken from a one-dimensional stellar model. We first identify several key factors that the one-dimensional initial model must satisfy to ensure efficient simulation of the convection process. We then use the three-dimensional simulations to examine the effects of two common modeling choices on the resulting convective flow: using a fixed composition approximation and using a reduced domain size. We find that using a fixed composition model actually increases the computational cost relative to using the full multi-species model because the fixed composition system takes longer to reach convection that is in a quasi-static state. Using a reduced (octant rather than full sphere) simulation domain yields flow with statistical properties that are within a factor of two of the full sphere simulation values. Both the octant and full sphere simulations show similar mixing across the convection zone boundary that is consistent with the turbulent entrainment model. However, the global character of the flow is distinctly different in the octant simulation, showing more rapid changes in the large-scale structure of the flow and thus a more isotropic flow on average.