Solution methods for the compressible Navier-Stokes equations based on finite volume discretizations often implement boundary conditions using ghost cells outside of the computational domain. Filling the ghost cells using straightforward zeroth-or first-order extrapolation, although computationally expedient, is well known to fail even for some simple flows, especially when turbulent structures interact with the boundaries or if time-varying inflow conditions are imposed. The Navier-Stokes characteristic boundary condition approach provides more accurate boundary conditions, but requires the use of special discretizations at boundaries. The present paper develops a new technique based on the Navier-Stokes characteristic boundary condition approach to derive values for ghost cells that significantly improve the treatment of boundaries over simple extrapolation, but retain the ghost cell approach. It is demonstrated in the context of a Godunov integration procedure that the new method provides accurate results, while allowing the use of the same stencil and numerical methodology near the boundaries as in the interior.

A low Mach number model for moist atmospheric flows is introduced that accurately incorporates reversible moist processes in flows whose features of interest occur on advective rather than acoustic time scales. Total water is used as a prognostic variable, so that water vapor and liquid water are diagnostically recovered as needed from an exact Clausius-Clapeyron formula for moist thermodynamics. Low Mach number models can be computationally more efficient than a fully compressible model, but the low Mach number formulation introduces additional mathematical and computational complexity because of the divergence constraint imposed on the velocity field. Here, latent heat release is accounted for in the source term of the constraint by estimating the rate of phase change based on the time variation of saturated water vapor subject to the thermodynamic equilibrium constraint. The authors numerically assess the validity of the low Mach number approximation for moist atmospheric flows by contrasting the low Mach number solution to reference solutions computed with a fully compressible formulation for a variety of test problems.

We present MAESTROeX, a massively parallel solver for low Mach number astrophysical flows. The underlying low Mach number equation set allows for efficient, long-time integration for highly subsonic flows compared to compressible approaches. MAESTROeX is suitable for modeling full spherical stars as well as well as planar simulations of dynamics within localized regions of a star, and can robustly handle several orders of magnitude of density and pressure stratification. Previously, we have described the development of the predecessor of MAESTROeX, called MAESTRO, in a series of papers. Here, we present a new, greatly simplified temporal integration scheme that retains the same order of accuracy as our previous approaches. We also explore the use of alternative spatial mapping of the one-dimensional base state onto the full Cartesian grid. The code leverages the new AMReX software framework for block-structured adaptive mesh refinement (AMR) applications, allowing for scalability to large fractions of leadership-class machines. Using our previous studies on the convective phase of single-degenerate progenitor models of SNe Ia as a guide, we characterize the performance of the code and validate the new algorithmic features. Like MAESTRO, MAESTROeX is fully open source.

Flows in which the primary features of interest do not rely on high-frequency acoustic effects, but in which long-wavelength acoustics play a nontrivial role, present a computational challenge. Integrating the entire domain with low-Mach-number methods would remove all acoustic wave propagation, while integrating the entire domain with the fully compressible equations can in some cases be prohibitively expensive due to the CFL time step constraint. For example, simulation of thermoacoustic instabilities might require fine resolution of the fluid/chemistry interaction but not require fine resolution of acoustic effects, yet one does not want to neglect the long-wavelength wave propagation and its interaction with the larger domain. The present paper introduces a new multi-level hybrid algorithm to address these types of phenomena. In this new approach, the fully compressible Euler equations are solved on the entire domain, potentially with local refinement, while their low-Mach-number counterparts are solved on subregions of the domain with higher spatial resolution. The finest of the compressible levels communicates inhomogeneous divergence constraints to the coarsest of the low-Mach-number levels, allowing the low-Mach-number levels to retain the long-wavelength acoustics. The performance of the hybrid method is shown for a series of test cases, including results from a simulation of the aeroacoustic propagation generated from a Kelvin–Helmholtz instability in low-Mach-number mixing layers. It is demonstrated that compared to a purely compressible approach, the hybrid method allows time-steps two orders of magnitude larger at the finest level, leading to an overall reduction of the computational time by a factor of 8.

The gas in galaxy clusters is heated by shock compression through accretion (outer shocks) and mergers (inner shocks). These processes additionally produce turbulence. To analyse the relation between the thermal and turbulent energies of the gas under the influence of non-adiabatic processes, we performed numerical simulations of cosmic structure formation in a box of 152 Mpc comoving size with radiative cooling, UV background, and a subgrid scale model for numerically unresolved turbulence. By smoothing the gas velocities with an adaptive Kalman filter, we are able to estimate bulk flows towards cluster cores. This enables us to infer the velocity dispersion associated with the turbulent fluctuation relative to the bulk flow. For haloes with masses above 1013 M⊙, we find that the turbulent velocity dispersions averaged over the warm-hot intergalactic medium (WHIM) and the intracluster medium (ICM) are approximately given by powers of the mean gas temperatures with exponents around 0.5, corresponding to a roughly linear relation between turbulent and thermal energies and transonic Mach numbers. However, turbulence is only weakly correlated with the halo mass. Since the power-law relation is stiffer for the WHIM, the turbulent Mach number tends to increase with the mean temperature of the WHIM. This can be attributed to enhanced turbulence production relative to dissipation in particularly hot and turbulent clusters.

We present a computational framework for solving the equations of inviscid
gas dynamics using structured grids with embedded geometries. The novelty of
the proposed approach is the use of high-order discontinuous Galerkin (dG)
schemes and a shock-capturing Finite Volume (FV) scheme coupled via an $hp$
adaptive mesh refinement ($hp$-AMR) strategy that offers high-order accurate
resolution of the embedded geometries. The $hp$-AMR strategy is based on a
multi-level block-structured domain partition in which each level is
represented by block-structured Cartesian grids and the embedded geometry is
represented implicitly by a level set function. The intersection of the
embedded geometry with the grids produces the implicitly-defined mesh that
consists of a collection of regular rectangular cells plus a relatively small
number of irregular curved elements in the vicinity of the embedded boundaries.
High-order quadrature rules for implicitly-defined domains enable high-order
accuracy resolution of the curved elements with a cell-merging strategy to
address the small-cell problem. The $hp$-AMR algorithm treats the system with a
second-order finite volume scheme at the finest level to dynamically track the
evolution of solution discontinuities while using dG schemes at coarser levels
to provide high-order accuracy in smooth regions of the flow. On the dG levels,
the methodology supports different orders of basis functions on different
levels. The space-discretized governing equations are then advanced explicitly
in time using high-order Runge-Kutta algorithms. Numerical tests are presented
for two-dimensional and three-dimensional problems involving an ideal gas. The
results are compared with both analytical solutions and experimental
observations and demonstrate that the framework provides high-order accuracy
for smooth flows and accurately captures solution discontinuities.

The convective period leading up to a Type Ia supernova (SN Ia) explosion is characterized by very low Mach number flows, requiring hydrodynamical methods well-suited to long-time integration. We continue the development of the low Mach number equation set for stellar scale flows by incorporating the effects of heat release due to external sources. Low Mach number hydrodynamics equations with a time-dependent background state are derived, and a numerical method based on the approximate projection formalism is presented. We demonstrate through validation with a fully compressible hydrodynamics code that this low Mach number model accurately captures the expansion of the stellar atmosphere as well as the local dynamics due to external heat sources. This algorithm provides the basis for an efficient simulation tool for studying the ignition of SNe Ia.

The dynamics of helium shell convection driven by nuclear burning establish the conditions for runaway in the sub-Chandrasekhar-mass, double-detonation model for SNe Ia, as well as for a variety of other explosive phenomena. We explore these convection dynamics for a range of white dwarf core and helium shell masses in three dimensions using the low Mach number hydrodynamics code MAESTRO. We present calculations of the bulk properties of this evolution, including time-series evolution of global diagnostics, lateral averages of the 3D state, and the global 3D state. We find a variety of outcomes, including quasi-equilibrium, localized runaway, and convective runaway. Our results suggest that the double-detonation progenitor model is promising and that 3D dynamic convection plays a key role.