MAESTRO Framework for Stellar Convection Simulations

Updated 16 August 2025

MAESTRO Framework is a computational tool that uses a low Mach number formulation to simulate slow, convection-dominated stellar flows.
It couples a dynamic hydrostatic base state with adaptive mesh refinement to resolve steep gradients in astrophysical convection.
The framework achieves significant speedups and accuracy, making it vital for studying pre-supernova convective processes in stars.

MAESTRO is a computational framework for simulating long-duration, highly subsonic, convection-dominated flows in stellar interiors, with particular emphasis on the convective phase preceding Type Ia supernova ignition. By employing a low Mach number hydrodynamics formulation and coupling it to adaptive mesh refinement (AMR), MAESTRO enables high-fidelity and efficient modeling of slow astrophysical flows that would be computationally prohibitive for traditional compressible codes. Its distinguishing features include the use of a time-dependent, hydrostatic base state, a multidimensional–base state mapping procedure, and domain-decomposed AMR infrastructure, positioning MAESTRO as an essential tool for tackling the computational challenges inherent in stellar convection studies (Nonaka et al., 2010).

1. Low Mach Number Formulation and Equation Set

MAESTRO is built upon a low Mach number approximation, where the full compressible Euler equations are asymptotically reduced so that fast acoustic phenomena are filtered out. This is achieved by decomposing the pressure field into a one-dimensional hydrostatic "base state" ( $p_0$ ) and a small dynamic perturbation. The governing equations ensure that the base state evolves on thermodynamic time scales, while the full fluid solution on the multidimensional grid evolves with acoustic waves suppressed. A crucial constraint equation is imposed:

$\nabla \cdot (\beta_0 \mathbf{u}) = \beta_0 \left(S - \frac{1}{p_0} \frac{\partial p_0}{\partial t}\right)$

$\beta_0$ is a density-like variable, typically related to the base state density $\rho_0$ , $p_0$ , and a thermodynamic derivative such as an average of $\Gamma_1$ .
$\mathbf{u}$ is the full velocity, decomposed into a base state radial velocity $w_0$ and local velocity fluctuations.
$S$ represents local compressive sources from heating, compositional evolution, or chemical reactions.

The evolution of the base state is governed by

$\frac{\partial \rho_0}{\partial t} + \nabla \cdot (\rho_0 w_0 \mathbf{e}_r) = 0$

$\frac{\partial p_0}{\partial r} = -\rho_0 g$

where $g$ is the gravitational acceleration, and $\mathbf{e}_r$ is the radial unit vector. This splitting isolates the slow evolution of stratification and thermodynamic state from fast, uninteresting acoustics, letting the code take time steps limited only by the fluid velocity rather than the sound speed.

MAESTRO uses a domain-decomposed, block-structured AMR methodology to manage spatial resolution:

The computational volume is initially covered by a coarse base grid, with finer nested grids dynamically added in regions exhibiting strong gradients or active flow (e.g., rising hot bubbles).
Synchronization between refinement levels is maintained by ghost cell filling from parent grids and conservative flux corrections at coarse–fine interfaces.
Special handling of the 1D base state ensures its synchronous evolution with the multidimensional grid: in planar geometry, the base state aligns with the grid; in spherical geometry, the base state employs a fixed radial resolution, with averages restricted to refined subregions that fully cover a given radial shell.

This approach allows substantial computational savings—adaptive grids yield speedups of 2–3× relative to uniform high-resolution runs—while maintaining second-order accuracy for key variables.

3. Multidimensional–Base State Mapping

Consistent coupling between the multidimensional solution and the one-dimensional base state is essential for thermodynamic consistency. MAESTRO uses two principal techniques:

Lateral Averaging: Extracts effective radial profiles by horizontally averaging the 3D scalar fields (such as $\rho$ or enthalpy) at each radial position. In spherical geometry, this requires quadrature and quadratic interpolation to compensate for non-alignment and sparse bin sampling.
"Fill" Operations: Projects 1D base state quantities (e.g., $p_0$ , $\rho_0$ ) onto Cartesian grid points via direct assignment (when precisely aligned) or high-order interpolation (e.g., a fourth-order formula in planar geometry):

$\phi_{i,j+1} = \frac{7}{12}(\phi_j + \phi_{j+1}) - \frac{1}{12}(\phi_{j-1} + \phi_{j+2})$

with monotonicity constraints imposed to prevent the appearance of spurious minima or maxima.

This strict two-way mapping ensures fidelity between layer-averaged state variables and multidimensional fluctuations, preserving hydrostatic and thermodynamic integrity.

4. Computational Efficiency, Accuracy, and Applications

Eliminating the acoustic time step restriction translates into significantly larger simulation time steps—often by a factor of $1/M$ (where $M$ is the Mach number)—over compressible codes. Coupled with AMR, this yields:

Demonstrated speedups of 2–3× in full-star convection simulations, and dramatic reductions in total cell count required.
Proven second-order convergence in both space and time for critical variables ( $\rho$ , $u$ , $h$ ).
Suitability for long-integration simulations of stellar convection, e.g., the simulation of rising convective plumes and turbulent pre-ignition flows in the convective phase of Type Ia supernovae.

These properties make MAESTRO particularly well-suited for resolving the multi-scale, turbulent convective structures leading up to supernova ignition, within feasible computational resource budgets.

5. AMR–Base State Coupling and Spherical Geometry Considerations

A major distinction from prior AMR implementations for incompressible flows is the time- and solution-dependent 1D base state, whose evolution is not independent but tightly coupled with the full 3D (or 2D) domain. Special care in mapping (both in lateral averaging and fill operations) is necessary:

In spherical geometry, grid cells are not radially aligned. The algorithm averages only over regions that cover an entire radial bin. Interpolation is used where direct coverage is insufficient.
Fixed radial base state resolution across levels ensures that measurement of global quantities such as gravitational acceleration and stratification remain consistent, independent of local refinement.

This assures fidelity and avoids introducing artifacts due to the misalignment of the multidimensional mesh and the stratification direction.

6. Limitations, Future Directions, and Theoretical Significance

While highly efficient for subsonic, convection-dominated modes, the low Mach number formalism precludes modeling of strong shocks and fast compressible phenomena, making MAESTRO unsuitable for the post-ignition (explosive) phase of Type Ia supernovae. Extensions such as coupling to fully compressible codes (e.g., CASTRO) are needed for seamless end-to-end simulation (see (Nonaka et al., 2011)).

Further advances are suggested:

Improved mapping techniques and error control for low "hit count" radial bins, especially in aspherical and highly turbulent cases.
Integration with massively parallel AMR frameworks to scale to even larger problem sizes.
Application to broader classes of stratified subsonic flows in stellar astrophysics, including x-ray bursts and convection in massive stars (Zingale et al., 2017).

MAESTRO thus marks a foundational advance in the computational modeling of low Mach number astrophysical flows, uniting asymptotic analysis, efficient AMR infrastructure, and sophisticated multidimensional–base state coupling for transformative increases in simulation fidelity and reach.