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.Comment: Accepted to Astrophysical Journal Suppliment (http://iop.org). 56 pages, 15 figures

A Hybrid Adaptive Low-Mach-Number/Compressible Method: Euler Equations

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

Low Mach Number Modeling of Type Ia Supernovae. IV. White Dwarf Convection

We present the first three-dimensional, full-star simulations of convection in a white dwarf preceding a Type Ia supernova, specifically the last few hours before ignition. For these long-time calculations we use our low Mach number hydrodynamics code, MAESTRO, which we have further developed to treat spherical stars centered in a three-dimensional Cartesian geometry. The main change required is a procedure to map the one-dimensional radial base state to and from the Cartesian grid. Our models recover the dipole structure of the flow seen in previous calculations, but our long-time integration shows that the orientation of the dipole changes with time. Furthermore, we show the development of gravity waves in the outer, stable portion of the star. Finally, we evolve several calculations to the point of ignition and discuss the range of ignition radii.Comment: 42 pages, some figures degraded to conserve space. Accepted to The Astrophysical Journal (http://journals.iop.org/

Multidimensional Modeling of Type I X-ray Bursts. I. Two-Dimensional Convection Prior to the Outburst of a Pure Helium 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.Comment: 47 pages including 18 figures; submitted to ApJ; A version with higher resolution figures can be found at http://astro.sunysb.edu/cmalone/research/pure_he4_xrb/ms.pd

A Numerical Study of Methods for Moist Atmospheric Flows: Compressible Equations

We investigate two common numerical techniques for integrating reversible moist processes in atmospheric flows in the context of solving the fully compressible Euler equations. The first is a one-step, coupled technique based on using appropriate invariant variables such that terms resulting from phase change are eliminated in the governing equations. In the second approach, which is a two-step scheme, separate transport equations for liquid water and vapor water are used, and no conversion between water vapor and liquid water is allowed in the first step, while in the second step a saturation adjustment procedure is performed that correctly allocates the water into its two phases based on the Clausius-Clapeyron formula. The numerical techniques we describe are first validated by comparing to a well-established benchmark problem. Particular attention is then paid to the effect of changing the time scale at which the moist variables are adjusted to the saturation requirements in two different variations of the two-step scheme. This study is motivated by the fact that when acoustic modes are integrated separately in time (neglecting phase change related phenomena), or when sound-proof equations are integrated, the time scale for imposing saturation adjustment is typically much larger than the numerical one related to the acoustics

Conservative Initial Mapping For Multidimensional Simulations of Stellar Explosions

Mapping one-dimensional stellar profiles onto multidimensional grids as initial conditions for hydrodynamics calculations can lead to numerical artifacts, one of the most severe of which is the violation of conservation laws for physical quantities such as energy and mass. Here we introduce a numerical scheme for mapping one-dimensional spherically-symmetric data onto multidimensional meshes so that these physical quantities are conserved. We validate our scheme by porting a realistic 1D Lagrangian stellar profile to the new multidimensional Eulerian hydro code CASTRO. Our results show that all important features in the profiles are reproduced on the new grid and that conservation laws are enforced at all resolutions after mapping.Comment: 7 pages, 5 figures, Proceeding for Conference on Computational Physics (CCP 2011

Low Mach Number Modeling of Type Ia Supernovae

We introduce a low Mach number equation set for the large-scale numerical simulation of carbon-oxygen white dwarfs experiencing a thermonuclear deflagration. Since most of the interesting physics in a Type Ia supernova transpires at Mach numbers from 0.01 to 0.1, such an approach enables both a considerable increase in accuracy and savings in computer time compared with frequently used compressible codes. Our equation set is derived from the fully compressible equations using low Mach number asymptotics, but without any restriction on the size of perturbations in density or temperature. Comparisons with simulations that use the fully compressible equations validate the low Mach number model in regimes where both are applicable. Comparisons to simulations based on the more traditional anelastic approximation also demonstrate the agreement of these models in the regime for which the anelastic approximation is valid. For low Mach number flows with potentially finite amplitude variations in density and temperature, the low Mach number model overcomes the limitations of each of the more traditional models and can serve as the basis for an accurate and efficient simulation tool.Comment: Accepted for publication in the Astrophysical Journal 31 pages, 5 figures (some figures degraded in quality to conserve space