MAESTROeX: A Massively Parallel Low Mach Number Astrophysical Solver

Alice Harpole; Andrew Nonaka; Ann S. Almgren; Duoming Fan; Michael Zingale

REVIEW 4 major objections 6 minor 1 cited by

MAESTROeX: A Massively Parallel Low Mach Number Astrophysical Solver

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A simplified temporal integration scheme for low Mach number astrophysical flows eliminates base-state evolution equations while retaining second-order accuracy, and an irregularly spaced base state nearly eliminates interpolation error.

desk verdict A genuinely simpler low Mach number astrophysical solver with real scaling results, but the second-order accuracy claim is unsupported by any convergence study and the irregular base state shows unexplained sensitivity. read the letter →

arxiv 1908.03634 v1 pith:HEMN4K4D submitted 2019-08-09 physics.comp-ph astro-ph.SR

Duoming Fan , Andrew Nonaka , Ann S. Almgren , Alice Harpole , Michael Zingale This is my paper · ORCID

classification physics.comp-phastro-ph.SR

keywords lowMachnumberhydrodynamicsastrophysicalconvectionstratifiedbasestateadaptivemeshrefinementmassivelyparallelcomputingwhitedwarfTypeIasupernovae

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

MAESTROeX is a solver for astrophysical flows that are slow enough that sound waves carry negligible energy, allowing much longer time steps than compressible hydrodynamics. The paper's central claim is that the temporal integration scheme can be greatly simplified without losing accuracy: scalar and velocity advection use the full velocity, velocity splitting is reserved for enforcing the divergence constraint at the star's boundary, and the one-dimensional hydrostatic base state is updated by a predictor-corrector from lateral averages instead of its own evolution equations. The paper reports that this preserves second-order accuracy in time, and validates it on convection preceding ignition in a white dwarf. It also introduces an irregularly spaced radial base state for spherical stars that nearly eliminates interpolation error in mapping between the one-dimensional background and the three-dimensional grid, at some added computational cost. If these claims are right, long-time convective simulations of full stars become both simpler and substantially more practical at scale.

What carries the argument

The load-bearing mechanism is the pair consisting of the one-dimensional hydrostatic base state $(\rho_0, p_0)$ and the divergence constraint on velocity, $\nabla \cdot (\beta_0 U) = \beta_0\left(S - \frac{1}{\Gamma_1 p_0}\frac{\partial p_0}{\partial t}\right)$, which replaces sound-wave dynamics in the low Mach number limit. The base state carries the stratification and enters the momentum equation through buoyancy; $\beta_0$ is a density-like variable carrying background stratification. The new algorithm computes the base state update by lateral averaging and one-dimensional hydrostatic integration instead of evolution equations, and uses the full velocity for advection, retaining the split into a radial base-state velocity and perturbation only in the projection step to enforce the boundary condition at the cutoff density at the star's edge.

What would settle it

Take a smooth stratified test problem with a manufactured solution and halve the time step: if the error does not decrease by roughly a factor of four, the claim of retained second-order temporal accuracy is false.

Watch

Extended reading notes

Core claim

At the core of the algorithm is the low Mach number equation set for a stratified star, in which total pressure is written $p = p_0(r,t) + \pi(x,t)$ with the dynamic part $\pi$ satisfying $|\pi|/p_0 = O(Ma^2)$, and the equations are closed by the divergence constraint $\nabla \cdot (\beta_0 U) = \beta_0\left(S - \frac{1}{\Gamma_1 p_0}\frac{\partial p_0}{\partial t}\right)$. The paper claims that the temporal integration of this system can be restructured so that the base state density and pressure are obtained at the new time by laterally averaging the updated Cartesian density and integrating hydrostatic equilibrium, rather than by solving evolution equations for the base state. Advection of scalars and velocity uses the full velocity field, and the earlier split into a radial base-state velocity plus perturbation is kept only to enforce the correct boundary behavior near the cutoff density at the star's edge. In white-dwarf convection tests through ignition, the new scheme agrees with the previous algorithm's qualitative evolution and predicts ignition at $t = 7810$ s versus $t = 7850$ s, while running about 13% faster per time step; the paper takes this as evidence that the simplification retains the same overall second-order accuracy.

Load-bearing premise

The whole method assumes the flow stays highly subsonic with pressure nearly in hydrostatic balance along one radial direction, so the dynamic pressure is tiny compared with the background pressure; where sound waves or strong lateral pressure variations matter, the simplified equations no longer represent the physics.

Editorial extensions

If this is right

Time steps are limited by the advective CFL condition rather than the sound speed, so highly subsonic convection can be integrated over long physical times at a fraction of the cost of compressible approaches.
The simplified temporal scheme retains second-order accuracy while removing base-state evolution equations, and is more amenable to higher-order multiphysics coupling strategies such as deferred-correction time integrators.
The irregularly spaced base state nearly eliminates interpolation error in spherical mapping, keeping a hydrostatic star essentially at rest when reactions are off; its cost is a larger radial array and about 25% more runtime per step, which could be reduced by truncating beyond the cutoff density.
Adaptive mesh refinement with two levels reproduces the dynamics of a single-level $512^3$ white-dwarf run at more than a factor of two in speedup.
Weak scaling holds to more than 10,000 MPI processes, with the lateral-average global reduction identified as the main scaling bottleneck rather than the linear solvers.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next test, which the paper lists as future work but does not demonstrate, is to wrap this simplified stepping scheme in spectral deferred corrections; the expected payoff is higher-order accuracy in the reactive coupling, but stability and convergence are unproven here.
Because the paper identifies the lateral-average global reduction as the main scaling bottleneck, a testable extension is to replace it with hierarchical or approximate parallel reductions and measure whether weak scaling improves without changing the solution beyond solver tolerance.
The irregular base state's early-time sensitivity to a coarse initial model near the star's center suggests a practical rule: the initial model's radial resolution near the center should be matched to the irregular spacing before drawing physical conclusions from ignition-time differences.
A boundary of validity follows from the pressure-split assumption: in regimes with strong lateral pressure gradients or propagating acoustic disturbances, a comparison against a compressible solver would delimit where the simplified base-state coupling is trustworthy.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Mike, here's my read on the MAESTROeX paper.

The real news is the simplified temporal integrator. The old MAESTRO split the velocity into base-state and perturbational parts and evolved the base state through its own PDEs. Here they advect with the full velocity, reconstruct the base state density by lateral averaging and the pressure by enforcing hydrostatic equilibrium, and use a predictor-corrector structure. That is genuinely simpler, and removing the base-state evolution equations cuts time per step by about 13% and, more importantly, removes an entire class of algorithmic complexity. The irregularly-spaced base state for spherical geometry is also a reasonable idea: mapping errors between the Cartesian grid and a uniform radial grid can generate spurious buoyancy, and eliminating interpolation is worth having.

The scaling results are credible. Weak scaling to 13,824 processes with only 42% wall-clock increase per step is decent, and the observation that the average operator, not the linear solvers, dominates the scaling is useful for the community. The code is open source, which is a big plus.

Now the soft spots. The paper claims the new scheme "retains the same overall second-order accuracy in time" as the previous approach, but there is no convergence study anywhere. The validation is single trials of a chaotic convective flow, compared by eye through peak temperature and peak Mach number. The text itself says the curves are not expected to match exactly. That is not a demonstration of second-order accuracy. Worse, the time derivative of the base-state pressure appears in the divergence constraint at both the predictor and corrector stages (Eqs. 15, 25, 33), and the way ∂p0/∂t is estimated is only sketched in Appendix A via ηρ. On reading the steps, it is not clear the reconstructed p0 is time-centered to second order. If that reconstruction is only first-order, the main algorithmic contribution reduces to a faster first-order solver.

The irregular base state also shows a disturbing sensitivity: at 256^3, the ignition time changes by about 14% (6840 s vs 7810 s), attributed to the initial model resolution near the center. That attribution is plausible but not demonstrated. By 512^3 the first 1000 s agree better, but they never run the 512^3 case to ignition.

This is not a takedown. The algorithm is well described, the code ships, and the simplification is real. But the accuracy claim is load-bearing and unproven. A serious referee should ask for a convergence study in a simpler test problem (e.g., a rising bubble or a 2D cylindrical case) showing second-order behavior of the new base-state reconstruction, and ideally a resolution study for the chaotic convection.

Bottom line: worth sending to peer review, but the second-order claim should not ride on the reputation of Papers I–V. Who benefits? Computational astrophysicists working on subsonic flows, especially Type Ia progenitors and X-ray bursts. If I were the editor, I would send it out with a request for a convergence study before acceptance.

Referee Report

4 major / 6 minor

Summary. This paper presents MAESTROeX, a new AMReX-based implementation of the MAESTRO low Mach number solver for stratified, highly subsonic astrophysical flows, together with two algorithmic developments. First, the temporal integration scheme of the earlier MAESTRO series (Papers I-V) is simplified: full velocities are used for scalar and velocity advection, and the base-state density and pressure are no longer advanced by their own evolution equations but are reconstructed after each advection step by lateral averaging followed by a one-dimensional hydrostatic equilibrium solve, with the time derivative of the base-state pressure entering the divergence constraint estimated through the auxiliary eta-rho quantity described in Appendix A. The paper claims this simplification retains second-order accuracy in time. Second, an irregularly-spaced radial base state is introduced for spherical problems to reduce interpolation error in the fill and average mappings. The paper validates the new implementation on full-star white dwarf convection problems by comparing peak temperature and peak Mach number with the original MAESTRO algorithm and with the uniformly-spaced base state, reports weak scaling on up to 1536^3 cells (13,824 MPI processes) on Cori KNL, and demonstrates AMR in planar and spherical configurations. The code is released as open source.

Significance. If the claims hold, this is a useful methods contribution to low Mach number astrophysical fluid dynamics. The temporal integration scheme is a genuine algorithmic simplification that removes the base-state evolution machinery of Papers II-V, and the paper's engineering evidence is solid: open-source release with versioned dependencies (AMReX and Microphysics cited with DOIs/version identifiers), weak scaling to 13,824 MPI processes with only a 42% wallclock increase from smallest to largest run, and an honest diagnosis of the lateral-averaging operator, not the linear solvers, as the scaling bottleneck (Section 4.1). The regression check against the previously validated MAESTRO (ignition at 7810 s versus 7850 s for uniform spacing) is a sensible consistency test, and the paper is transparent about the limitations of the irregular base state: it is incompatible with AMR in spherical geometry (end of Section 3.2), costs about 25% more wallclock time (Section 4.2), and produces a 12% difference in ignition time whose cause is offered only as a conjecture (Section 4.2).

major comments (4)

[Section 3.2; Eqs. (15), (25), (33); Appendix A] The central claim that the new temporal scheme retains the same overall second-order accuracy in time is not supported by any convergence or truncation-error analysis. In the new scheme p0 is reconstructed rather than evolved (Steps 4B/C and 8B/C), yet its time derivative enters the divergence constraint at every projection (Eqs. 15, 25, 33), and that derivative is estimated through the laterally averaged eta-rho quantity described only in Appendix A (Eqs. A4-A6). The text justifies the accuracy claim by characterizing Steps 6-9 as a trapezoidal corrector, but it does not show that the reconstructed p0 and the eta-rho-based partial p0/partial t are time-centered to second order. As an example of the time-level mixing that is left unanalyzed, the constraint in Step 3 (Eq. 15) uses beta0^n and (partial p0/partial t)^{n-1/2} to construct a velocity intended to be time-centered at n+1/2. If a first-order imbalance in the predictor is exactly compensated by the corrector, that mechanism needs to be stated and verified; a convergence study with either a manufactured solution or a well-resolved reference solution would settle the issue. As written, the manuscript would not allow a reader to distinguish a second-order scheme from a first-order one on the evidence presented.
[Section 4.2, Figures 4-5] The validation in Section 4.2 does not provide the missing accuracy test. The diagnostics are peak temperature and peak Mach number from single realizations of a chaotic convective flow, and Section 4.3 explicitly disclaims exact agreement between runs. No error norm, Richardson extrapolation, or third resolution is reported; Figure 5 covers only the first 1000 s of a run that reaches ignition near 7000 s, and the text notes that 'limited allocations' prevented a longer run. Such qualitative agreement would be consistent with a first-order scheme and therefore cannot carry the paper's central claim. I recommend a convergence study on the planar hot-bubble problem used in Section 4.3, with errors measured against a high-resolution reference solution, together with a demonstration that the reconstructed base state and the projection constraints converge at the claimed rate as Delta t and Delta x are reduced together.
[Section 4.2] The comparison of the uniform- and irregularly-spaced base-state runs is confounded: the two simulations differ both in the mapping and in the initial model file, since 'the resolution near the center of the star is much coarser with the irregular spacing,' which the authors 'strongly suspect' causes the 6840 s versus 7810 s difference in ignition time. The ignition time is the principal scientific output of this problem, and a roughly 12% difference should not rest on a conjectural explanation. Please isolate the two effects, for example by interpolating a common initial model onto both base-state grids or by rerunning with an irregular initial model constructed consistently with the uniform one, and report the resolution dependence of the ignition time itself.
[Section 3.1; end of Section 3.2; Section 4.2] The paper motivates the irregularly-spaced base state by the static-star argument: in the absence of reactions a spherical star in hydrostatic equilibrium should remain at rest, and the new mapping should nearly eliminate the spurious buoyancy forcing from interpolation error. Yet the static-star rest test is never shown, and the actual reduction in mapping error is never quantified. Given the stated costs of the new mapping (incompatibility with AMR in spherical geometry, a roughly 25% wallclock increase from the larger base-state array), the benefit relative to these costs is not demonstrated. A simple rest-state experiment measuring spurious velocities as a function of resolution for the uniform and irregular mappings would directly test the motivating claim and would also exercise the HSE reconstruction at the heart of the new temporal scheme.

minor comments (6)

[Abstract; Section 1] There is a duplicated phrase 'as well as well as' in the abstract (it also appears in the introduction), and Section 1 contains a typo ('compressibilitiy'); the manuscript would benefit from a careful proofread.
[Section 1 vs. Section 3.2 and Appendix A] Section 1 states that a key design decision was to 'eliminate the splitting of the velocity into average and perturbational components,' but Section 3.2 and Appendix A (Eqs. A1-A3) show that the splitting is still used in every projection step, and Figure 9 demonstrates that it is needed for the nodal projection. Please reword the abstract and Section 1 to state the simplification accurately: full velocities are used for advection, while the velocity splitting is retained inside the projections.
[Figure 9] The caption and the in-text description of Figure 9 are hard to reconcile with the displayed panels; the caption describes the left panel as 'Velocity magnitude, solved using full U' while the surrounding text describes the left panel as a peak-temperature comparison. Please make the sub-figure labels and caption consistent with the panels.
[Section 3.1] The statement that the irregularly-spaced base-state spacing decreases with radius is counterintuitive at first reading; since the allowed radii are r_m = Delta x sqrt(0.75 + 2m) (Eq. 9), the spacing between consecutive radii decreases as Delta x / (2 sqrt(2m)) for large m. A sentence explaining this would help the reader connect the outward refinement to the larger array size and the cost increase reported in Section 4.2.
[Section 4.3] The remark that the code 'may encounter numerical difficulties when the tagging values are too close' between refinement levels is vague; please specify the failure mode and, if possible, give guidance on safe tagging-value ratios.
[Section 4.2, Figure 5] Please clarify whether the 256^3 curves in Figure 5 are the same simulations as those plotted in Figure 4; the differing axis ranges make this difficult to verify, and the relationship matters for interpreting the resolution study.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new temporal scheme and irregular base-state mapping are algorithmic developments, and no fitted parameter is relabeled as a prediction.

full rationale

MAESTROeX builds on the authors' own low Mach number equation set from Papers I-III, but those equations are cited as published derivations with stated asymptotic assumptions (|π|/p0 = O(Ma^2), Eq. 2 and Eq. 7), not re-derived from the present paper's outputs. The new temporal integrator (Section 3.2) replaces base-state evolution equations with a predictor-corrector reconstruction: ρ0 is defined as the lateral average of ρ (Average), and p0 is defined by the hydrostatic relation (Enforce HSE, Eq. 5). These are definitional algorithmic operations, not fitted parameters, and the paper does not claim to predict an independent quantity from them. The divergence constraints in Eqs. 15, 25, and 33 use ∂p0/∂t estimated via ηρ (Appendix A) from prior work; this is a closure approximation, not an equation that reduces to its own input. The retained-second-order-accuracy claim is asserted rather than demonstrated by a convergence study, and the validation compares single trials of the authors' own previous MAESTRO code; those are verification and reproducibility concerns, not evidence that a result is forced by construction. No equation in the paper reduces to another equation by construction in a way that makes a claimed prediction tautological, and no fitted value is renamed as a prediction.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the low Mach number approximation and the specific averaging and mapping choices; no new physical entities or fitted constants are introduced. The numerical parameters (CFL, cutoff density, tagging) are hand-chosen and affect the results but are not fitted to match a target.

free parameters (4)

CFL number (sigma_CFL) = ~0.7
Time step constraint in Eq. (13); chosen by hand as a typical value.
Cutoff density = rho ~ 10^5 g cm^-3 in white dwarf test
Density threshold outside which density is held constant to control edge velocities; chosen by hand (Section 3.2, Section 4.2).
Sponge parameters = not specified
Artificial damping of velocities near or beyond the cutoff region; parameters are mentioned but not quantified in Section 3.2.
AMR tagging thresholds = rho > 10^5 and rho > 10^8 g cm^-3
Tagging criteria for refinement in Section 4.3; chosen by hand.

assumptions (4)

domain assumption The total pressure is decomposed as p = p0(r,t) + pi(x,t) with |pi|/p0 = O(Ma^2), where p0 is hydrostatic (Eq. 5).
Foundation of the low Mach number model (Section 2); taken from Papers I-III.
domain assumption The EOS closure is enforced through the divergence constraint (Eq. 7), requiring that thermodynamic variables evolve consistent with p0.
Section 2; derived in Papers I-II.
domain assumption The low Mach number equation set (Eqs. 1-3) accurately describes reacting, stratified astrophysical flows.
Section 2; asymptotic derivation from compressible equations (Rehm & Baum 1978; Majda & Sethian 1985; Papers I-III).
domain assumption The base state density rho0 is the lateral average of rho and represents the background stratification.
Section 3.1; average operator defined in Section 3.1.

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Cite this review

Pith. "Pith review of MAESTROeX: A Massively Parallel Low Mach Number Astrophysical Solver." pith.science (2026). https://pith.science/paper/HEMN4K4D

@misc{pith2026190803634,
  author       = {Pith},
  title        = {Pith review of: MAESTROeX: A Massively Parallel Low Mach Number Astrophysical Solver},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HEMN4K4D}},
  note         = {Machine review of arXiv:1908.03634}
}

read the original abstract

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 Type Ia supernovae as a guide, we characterize the performance of the code and validate the new algorithmic features. Like MAESTRO, MAESTROeX is fully open source.

Figures

Figures reproduced from arXiv: 1908.03634 by the authors.

**Figure 1.** (Left) For multi-level problems in planar geometry, we force a direct alignment between the base state cell centers and the Cartesian grid cell centers by allowing the radial base state spacing to change with space and time. (Right) For multi-level problems in spherical geometry, since there is no direct alignment between the base state cell centers and the Cartesian grid cell centers, we choose to fix the radial ba… view at source ↗

**Figure 2.** A direct mapping between the base state cell centers (red squares) and the Cartesian grid cell centers (blue crosses) is enforced by computing the average of the grid cell centers that share the same radial distance from the center of the star [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** (Left) Weak scaling results for a spherical, full-star white dwarf calculation using the original MAESTRO code, MAESTROeX, and MAESTROeX with base state evolution disabled. Shown is the average wallclock time per time step. (Right) Weak scaling results showing the average wallclock time per time step spent in the cell-centered and nodal linear solvers within a full time step of the aforementioned simulations. In the… view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: (left) Peak temperature, Tpeak, and (right) peak Mach number in a white dwarf until time of ignition at resolution of 2563 for three different MAESTROeX algorithms. In terms of efficiency, all three simulations on 2563 single-level grid were run on Cori haswell with 64…

**Figure 5.** Figure 5: Peak temperature, Tpeak in a white dwarf at grid resolutions of 2563 (dotted line) and 5123 (solid line) until t = 1000 for uniform (dx) and irregular (dr) base state spacing. Note that the irregularly-spaced solution agrees better with the uniformly-spaced solution as…

**Figure 6.** Figure 6: Initial grid structures with two levels of refinement. The red, green, and black lines represent grids of increasing refinement. (Left) Profile of T −T¯ for a hot bubble in a white dwarf environment. (Right) Region of star where ρ ≥ 105 g cm−3 in a full white dwarf sta…

**Figure 7.** Figure 7: Time-lapse cross-section of a hot bubble in a white dwarf environment at t = 0, 1.25, and 2.5 s for (left) single-level simulation, and (right) adaptive simulation at the same effective resolution. The red, green, and black boxes indicate grids of increasing resolution…

**Figure 8.** Figure 8: Peak temperature, Tpeak, in a white dwarf from t = 0 to 2000 s for grids with effective resolution of 5123 . We can see that the adaptive grids with two levels of refinement give very similar solution trends compared to the single-level grid. (Almgren et al. 2019). We …

**Figure 9.** Figure 9: When solving for the full velocity U in the nodal projection step instead of the split velocity formulation, we illustrate (left) the peak temperature, Tpeak in a white dwarf at resolution of 2563 during the early evolution of the star and (right) the magnitude of velo…

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

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[2] [2]

write newline

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[3] [3]

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2017

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