Introduction to MAESTROeX

History of MAESTROeX

MAESTROeX models the evolution of low Mach number astrophysical flows that are in hydrostatic equilibrium. The name MAESTROeX itself derives from MAESTRO, the original (pure Fortran) low Mach number stellar hydrodynamics code. MAESTROeX began as a rewrite MAESTRO in the C++ framework of AMReX, and has since then gained additional algorithmic capabilities. Henceforth, we will refer only to MAESTROeX, although many of the ideas originated in MAESTRO before the change to MAESTROeX.

The idea for MAESTROeX grew out of our success in applying low Mach number combustion methods developed for terrestrial flames [DB00] to small-scale (non-stratified) astrophysical flames such as the Landau-Darrieus instability [BDR+04], Rayleigh-Taylor unstable flames [ZingaleWoosleyRendleman+05], and flame-turbulence interactions [AspdenBellDay+08]. Our original small-scale astrophysical combustion algorithm is detailed in

Adaptive Low Mach Number Simulations of Nuclear Flames, J. B. Bell, M. S. Day, C. A. Rendleman, S. E. Woosley, & M. Zingale 2004, JCP, 195, 2, 677 (henceforth BDRWZ) [BellDayRendleman+04]

MAESTROeX was developed initially for modeling the convective phase in a Chandrasekhar mass white dwarf preceding the ignition of a Type Ia supernovae. As such, we needed to incorporate the compressibility effects due to large-scale stratification in the star. The method closest in spirit to MAESTROeX is the pseudo-incompressible method of Durran [Dur89], developed for terrestrial atmospheric flows (assuming an ideal gas). Part of the complexity of the equations in MAESTROeX stems from the need to describe a general equation of state. Additionally, since reactions can significantly alter the hydrostatic structure of a star, we incorporated extensions that capture the expansion of the background state [Alm00]. The low Mach number equations for stellar flows were developed in a series of papers leading up to the first application to this problem:

Low Mach Number Modeling of Type Ia Supernovae. I. Hydrodynamics, A. S. Almgren, J. B. Bell, C. A. Rendleman, & M. Zingale 2006, ApJ, 637, 922 (henceforth paper I) [AlmgrenBellRendlemanZingale06a]
Low Mach Number Modeling of Type Ia Supernovae. II. Energy Evolution, A. S. Almgren, J. B. Bell, C. A. Rendleman, & M. Zingale 2006, ApJ, 649, 927 (henceforth paper II) [AlmgrenBellRendlemanZingale06b]
Low Mach Number Modeling of Type Ia Supernovae. III. Reactions, A. S. Almgren, J. B. Bell, A. Nonaka, & M. Zingale 2008, ApJ, 684, 449 (henceforth paper III) [AlmgrenBellNonakaZingale08]
Low Mach Number Modeling of Type Ia Supernovae. IV. White Dwarf Convection, M. Zingale, A. S. Almgren, J. B. Bell, A. Nonaka, & S. E. Woosley 2009, ApJ, 704, 196 (henceforth paper IV) [ZingaleAlmgrenBell+09]

The adaptive mesh refinement version of the algorithm was presented in the next papers in the series:

MAESTRO: An Adaptive Low Mach Number Hydrodynamics Algorithm for Stellar Flows, A. Nonaka, A. S. Almgren, J. B. Bell, M. J. Lijewski, C. M. Malone, & M. Zingale 2010, ApJS, 188, 358 (henceforth “the multilevel paper”) [NonakaAlmgrenBell+10]

The most recent developments for MAESTROeX are described in:

MAESTROeX: A Massively Parallel Low Mach Number Astrophysical Solver, D. Fan, A. Nonaka, A. S. Almgren, A. Harpole, & M. Zingale, 2019, submitted to ApJ [FanNonakaAlmgren+19]

We have many papers that describe applications of the method to Type Ia supernovae, X-ray bursts, and stellar evolution. These are listed on the main MAESTROeX website. Some of these papers have appendices that describe enhancements to the code—these are noted below.

Multidimensional Modeling of Type I X-ray Bursts. I. Two-dimensional Convection Prior to the Outburst of a Pure 4He Accretor, C. M. Malone, A. Nonaka, A. S. Almgren, J. B. Bell, & M. Zingale 2011, ApJ, 728, 118 (henceforth “the XRB paper”) [MaloneNonakaAlmgren+11]

This introduces the thermal diffusion portion of the MAESTROeX algorithm.
Comparisons of Two- and Three-Dimensional Convection in Type I X-ray Bursts M. Zingale, C. M. Malone, A. Nonaka, A. S. Almgren, & J. B. Bell 2015, ApJ, 807, 60. [ZingaleMaloneNonaka+15]

This has an appendix that describes the Godunov state construction in more detail than previous papers.
Low Mach Number Modeling of Convection in Helium Shells on Sub-Chandrasekhar White Dwarfs II: Bulk Properties of Simple Models, A. M. Jacobs, M. Zingale, A. Nonaka, A. S. Almgren, & J. B. Bell 2016, ApJ, 827, 84. [JacobsZingaleNonaka+16]

This has an appendix that shows some test problems for the alternate energy formulation in MAESTROeX.

Brief Overview of Low Speed Approximations

There are many low speed formulations of the equations of hydrodynamics in use, each with their own applications. All of these methods share in common a constraint equation on the velocity field that augments the equations of motion.

Incompressible Hydrodynamics

The simplest low Mach number approximation is incompressible hydrodynamics. This approximation is formally the zero Mach number limit (\(M \rightarrow 0\)) of the Navier-Stokes equations. In incompressible hydrodynamics, the velocity satisfies a constraint equation:

(1)\[\nabla \cdot \Ub = 0\]

which acts to instantaneously equilibrate the flow, thereby filtering out soundwaves. The constraint equation implies that

(2)\[D\rho/Dt = 0\]

(through the continuity equation) which says that the density is constant along particle paths. This means that there are no compressibility effects modeled in this approximation.

Anelastic Hydrodynamics

In the anelastic approximation small amplitude thermodynamic perturbations are carried with respect to a static hydrostatic background (described by density \(\rho_0\)). The density perturbation is ignored in the continuity equation, resulting in a constraint equation:

(3)\[\nabla \cdot (\rho_0 \Ub) = 0\]

This properly captures the compressibility effects due to the stratification of the background. Because there is no evolution equation for the perturbational density, approximations are made to the buoyancy term in the momentum equation.

Low-Mach Number Combustion

In the low Mach number combustion model, the pressure is decomposed into a dynamic, \(\pi\), and thermodynamic component, \(p_0\), the ratio of which is \(O(M^2)\). The total pressure is replaced everywhere by the thermodynamic pressure, except in the momentum equation. This decouples the pressure and density and filters out the sound waves. Large amplitude density and temperature fluctuations are allowed. The only requirement is that the total pressure stay close to the background pressure, which is assumed constant. This requirement can be expressed as:

(4)\[p = p_0\]

and differentiating this along particle paths leads to a constraint on the velocity field:

(5)\[\nabla \cdot \Ub = S\]

This looks like the constraint for incompressible hydrodynamics, but now we have a source term, \(S\), representing the local compressibility effects due to the energy generation and thermal diffusion. Since the background pressure is taken to be constant, we cannot model flows that cover a large fraction of a pressure scale height. However, this method is ideal for exploring the physics of flames.

Pseudo-Incompressible Methods

The pseudo-incompressible method incorporates both the local changes to compressibility due to reaction/heat release, and the large-scale changes due to the background stratification. This was originally derived for an ideal gas equation of state for atmospherical flows. Allowing the background pressure, \(p_0\) to vary (e.g. in hydrostatic equilibrium), differentiating pressure along particle paths gives:

(6)\[\nabla \cdot (p_0^{1/\gamma} \Ub) = H\]

where \(\gamma\) is the ratio of specific heats and \(H\) is the source.

MAESTROeX is based on this method, generalizing this constraint to an arbitrary equation of state and allowing for the time-variation of the base state.

Alternate Energy Formulation

Several authors [KP12, VLB+13] showed that with a slightly different momentum equation, the low Mach number system can conserve an energy (that is, a quantity that looks like the compressible energy, but formed using the low Mach number quantities). This change manifests itself as either a change to the buoyancy term or by changing \(\nabla \pi\) to \(\beta_0 \nabla (\pi/\beta_0)\). Furthermore, [VLB+13] showed that the new formulation better captures the vertical propagation of gravity waves. As of Dec. 2013, this new formulation is the default in MAESTROeX.

Projection Methods 101

Most astrophysical hydrodynamics codes (e.g. CASTRO [AlmgrenBecknerBell+10] or FLASH [FryxellOlsonRicker+00]) solve the compressible Euler equations, which can be written in the form:

(7)\[\Ub_t + \nabla \cdot F(\Ub) = 0\]

where \(\Ub\) is the vector of conserved quantities, \(\Ub = (\rho, \rho u, \rho E)\), with \(\rho\) the density, \(u\) the velocity, and \(E\) the total energy per unit mass. This system of equations can be expressed as a system of advection equations:

(8)\[{\bf q}_t + A({\bf q}) {\bf q}_x = 0\]

where \({\bf q}\) are called the primitive variables, and \(A\) is the Jacobian, \(A \equiv \partial F / \partial U\). The eigenvalues of the matrix \(A\) are the characteristic speeds—the speeds at which information propagates. For the Euler equations, these are \(u\) and \(u \pm c\), where \(c\) is the sound speed. Solution methods for the compressible equations make use of this wave-nature to compute fluxes at the interfaces of grid cells to update the state in time. An excellent introduction to these methods is provided by LeVeque’s book [LeVeque02]. The timestep for these methods is limited by the time it takes for the maximum characteristic speed to traverse one grid cell. For very subsonic flows, this means that the timestep is dominated by the propagation of soundwaves, which may not be important to the overall dynamics of the flow.

In contrast, solving low Mach number systems (including the equations of incompressible hydrodynamics) typically involves solving one or more advection-like equations (representing, e.g. conservation of mass and momentum) coupled with a divergence constraint on the velocity field. For example, the equations of constant-density incompressible flow are:

(9)\[\Ub_t = -\Ub \cdot \nabla \Ub + \nabla p\]

(10)\[\nabla \cdot \Ub = 0\]

Here, \(\Ub\) represents the velocity vector [1]_ and \(p\) is the dynamical pressure. The time-evolution equation for the velocity (Eq.9) can be solved using techniques similar to those developed for compressible hydrodynamics, updating the old velocity, \(\Ub^n\), to the new time-level, \(\Ub^\star\). Here the ‘\(^\star\)’ indicates that the updated velocity does not, in general, satisfy the divergence constraint. A projection method will take this updated velocity and force it to obey the constraint equation. The basic idea follows from the fact that any vector field can be expressed as the sum of a divergence-free quantity and the gradient of a scalar. For the velocity, we can write:

(11)\[\Ub^\star = \Ub^d + \nabla \phi\]

where \(\Ub^d\) is the divergence free portion of the velocity vector, \(\Ub^\star\), and \(\phi\) is a scalar. Taking the divergence of Eq.11, we have

(12)\[\nabla^2 \phi = \nabla \cdot \Ub^\star\]

(where we used \(\nabla \cdot \Ub^d = 0\)). With appropriate boundary conditions, this Poisson equation can be solved for \(\phi\), and the final, divergence-free velocity can be computed as

(13)\[\Ub^{n+1} = \Ub^\star - \nabla \phi\]

Because soundwaves are filtered, the timestep constraint now depends only on \(|\Ub|\).

Extensions to variable-density incompressible flows [BM92] involve a slightly different decomposition of the velocity field and, as a result, a slightly different Poisson equation. There is also a variety of different ways to express what is being projected [ABC00], and different discretizations of the divergence and gradient operators lead to slightly different mathematical properties of the methods (leading to “approximate projections” [ABS96]). Finally, for second-order methods, two projections are typically done per timestep. The first (the ‘MAC’ projection [BCH91]) operates on the half-time, edge-centered advective velocities, making sure that they satisfy the divergence constraint. These advective velocities are used to construct the fluxes through the interfaces to advance the solution to the new time. The second/final projection operates on the cell-centered velocities at the new time, again enforcing the divergence constraint. The MAESTROeX algorithm performs both of these projections.

The MAESTROeX algorithm builds upon these ideas, using a different velocity constraint equation that captures the compressibility due to local sources and large-scale stratification.