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DOE OSTI2025

Uncertainty Quantification Enabled by Automatic Differentiation for Hydrodynamic Simulation of Shock‐to‐Detonation Transition in High Explosives

Quantifying the effects of uncertainty in a reactive burn model on the run-to-detonation time in high explosives (HEs) provides a robust methodology for assessing the probability of an HE failing the IHE qualification standard. Moreover, uncertainty quantification helps evaluate whether the model calibration accurately represents data outside the calibration set. This study uses a specialized hydrodynamic simulation code for modeling detonation to determine the run-to-detonation time of the HE PBX 9502 for various impact velocities. To quickly approximate uncertainties in the model, a surrogate was constructed using a Taylor series expansion centered at the mean of the input parameters. To obtain the sensitivities required for constructing the Taylor series, HYP-percomplex Automatic Differentiation (HYPAD) was implemented. HYPAD is a methodology for infusing existing codes with automatic differentiation capabilities by augmenting variables with one or more imaginary units to compute step-size independent partial derivatives. These derivatives are accurate to machine precision with respect to the implemented numerical algorithm, meaning their accuracy reflects that of the underlying method (e.g., integration or discretization schemes). Using reduced order modeling techniques, the mean and standard deviation of the run-to-detonation time of a shock within PBX 9502 were computed for a number of initial impact velocities. A weighted least squares regression was then performed to obtain a best fit curve and prediction interval for the computed statistics. Historical data points from explosively driven wedge tests were utilized to validate the prediction interval, ensuring its reliability in predicting future outcomes. With this prediction interval and a known safety constraint curve, the most probable point of failure and the probability of failure for the HE PBX 9502 were determined.

97 MATHEMATICS AND COMPUTING↗

DOE OSTI2025

Testing the thermal Sunyaev-Zel’dovich power spectrum of a halo model using hydrodynamical simulations

Statistical properties of large-scale cosmological structures serve as powerful tools for constraining the cosmological properties of our Universe. Tracing the gas pressure, the thermal Sunyaev-Zel’dovich (tSZ) effect is a biased probe of mass distribution and, hence, can be used to test the physics of feedback or cosmological models. Therefore, it is crucial to develop robust modelling of hot gas pressure for applications to tSZ surveys. Since gas collapses into bound structures, it is expected that most of the tSZ signal is within halos produced by cosmic accretion shocks. Hence, simple empirical halo models can be used to predict the tSZ power spectra. In this study, we employed the HMx halo model to compare the tSZ power spectra with those of several hydrodynamical simulations: the Horizon suite and the Magneticum simulation. We examine various contributions to the tSZ power spectrum across different redshifts, including the one- and two-halo term decomposition, the amount of bound gas, the importance of different masses, and the electron pressure profiles. Our comparison of the tSZ power spectrum reveals discrepancies between the halo model and cosmological simulations that increase with redshift. We find a 20% to 50% difference between the measured and predicted tSZ angular power spectrum over the multipole range ℓ = 10 3 − 10 4 . Our analysis reveals that these differences are driven by the excess of power in the predicted two-halo term at low k and in the one-halo term at high k . At higher redshifts ( z ∼ 3), simulations indicate that more power comes from outside the virial radius than from inside, suggesting a limitation in the applicability of the halo model. We also observe differences in the pressure profiles, despite the fair level of agreement on the tSZ power spectrum at low redshift with the default calibration of the halo model. In conclusion, our study suggests that the properties of the halo model need to be carefully controlled against real or mock data to be proven useful for cosmological purposes.

Ayçoberry, Emma (ORCID:0000000292351195)↗

DOE OSTI2023

Simulating Hydrodynamics in Cosmology with CRK-HACC

Abstract We introduce CRK-HACC, an extension of the Hardware/Hybrid Accelerated Cosmology Code (HACC), to resolve gas hydrodynamics in large-scale structure formation simulations of the universe. The new framework couples the HACC gravitational N -body solver with a modern smoothed-particle hydrodynamics (SPH) approach called conservative reproducing kernel SPH (CRKSPH). CRKSPH utilizes smoothing functions that exactly interpolate linear fields while manifestly preserving conservation laws (momentum, mass, and energy). The CRKSPH method has been incorporated to accurately model baryonic effects in cosmology simulations—an important addition targeting the generation of precise synthetic sky predictions for upcoming observational surveys. CRK-HACC inherits the codesign strategies of the HACC solver and is built to run on modern GPU-accelerated supercomputers. In this work, we summarize the primary solver components and present a number of standard validation tests to demonstrate code accuracy, including idealized hydrodynamic and cosmological setups, as well as self-similarity measurements.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2023

Bayesian calibration of viscous anisotropic hydrodynamic simulations of heavy-ion collisions

Owing to large pressure gradients at early times, standard hydrodynamic model simulations of relativistic heavy-ion collisions do not become reliable until O(1) fm/c after the collision. To address this one often introduces a prehydrodynamic stage that models the early evolution microscopically, typically as a conformal, weakly interacting gas. In such an approach the transition from the prehydrodynamic to the hydrodynamic stage is discontinuous, introducing considerable theoretical model ambiguity. Alternatively, fluids with large anisotropic pressure gradients can be handled macroscopically using the recently developed viscous anisotropic hydrodynamics (VAH). In high-energy heavy-ion collisions VAH is applicable already at very early times, and at later times transitions smoothly into conventional second-order viscous hydrodynamics. Here we present a Bayesian calibration of the VAH model with experimental data for Pb-Pb collisions at the LHC at $\sqrt{^{S}NN}$ = 2.76 A TeV. We find that the VAH model has the unique capability of constraining the specific viscosities of the quark-gluon plasma at higher temperatures than other previously used models.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

NASA NTRS1979

Hydrodynamic simulations of flare/surge events

A one-dimensional, hydrodynamic, time-dependent model which simulates the major observed dynamics of flare associated surges was developed. The thermodynamics, the surge mass, the time scales, the physical dimensions, and the velocities of typical surge events are reproduced. The surge is created by a sudden pressure increase at the top of the chromosphere, and this pressure pulse produces a disturbance which is followed, with a time-dependent numerical solution, as it propagates upward through the transition region and into the corona. The leading edge of the disturbance is a weak shock which has only a slight effect on the original transition region and coronal thermodynamics. The major effect occurs behind the shock where the temperature is decreased and the density is increased, and this cool, dense region moves upward then falls downward. After the material begins returning to the chromosphere, a second shock is formed which propagates upward, brings the infalling material to rest, and returns the atmosphere to hydrostatic equilibrium.

Steinolfson, R. S.↗

NASA NTRS2013

Constraints on Decreases in Eta Carinae's Mass-loss from 3D Hydrodynamic Simulations of Its Binary Colliding Winds

Recent work suggests that the mass-loss rate of the primary star Eta-A in the massive colliding wind binary Eta Carinae dropped by a factor of 2-3 between 1999 and 2010. We present result from large- (+/- 1545 au) and small- (+/- 155 au) domain, 3D smoothed particle hydrodynamics (SPH) simulations of Eta Car's colliding winds for three Eta-A mass-loss rates ( (dot-M(sub Eta-A) = 2.4, 4.8 and 8.5 × 10(exp −4) M(solar)/ yr), investigating the effects on the dynamics of the binary wind-wind collision (WWC). These simulations include orbital motion, optically thin radiative cooling and radiative forces. We find that dot-M Eta-A greatly affects the time-dependent hydrodynamics at all spatial scales investigated. The simulations also show that the post-shock wind of the companion star Eta-B switches from the adiabatic to the radiative-cooling regime during periastron passage (Phi approx.= 0.985-1.02). This switchover starts later and ends earlier the lower the value of dot-M Eta-A and is caused by the encroachment of the wind of Eta-A into the acceleration zone of Eta-B's wind, plus radiative inhibition of Eta-B's wind by Eta-A. The SPH simulations together with 1D radiative transfer models of Eta-A's spectra reveal that a factor of 2 or more drop in dot-M EtaA should lead to substantial changes in numerous multiwavelength observables. Recent observations are not fully consistent with the model predictions, indicating that any drop in dot- M Eta-A was likely by a factor of approx. < 2 and occurred after 2004. We speculate that most of the recent observed changes in Eta Car are due to a small increase in the WWC opening angle that produces significant effects because our line of sight to the system lies close to the dense walls of the WWC zone. A modest decrease in dot-M Eta-A may be responsible, but changes in the wind/stellar parameter of Eta-B, while less likely, cannot yet be fully ruled out. We suggest observations during Eta-Car's next periastron in 2014 to further test for decreases in dot-M Eta-A. If dot-M Eta-A is declining and continues to do so, the 2014 X-ray minimum should be even shorter than that of 2009.

Eta Carinae↗

NASA NTRS1994

X-ray clusters in a cold dark matter + lambda universe: A direct, large-scale, high-resolution, hydrodynamic simulation

A new, three-dimensional, shock-capturing, hydrodynamic code is utilized to determine the distribution of hot gas in a cold dark matter (CDM) + lambda model universe. Periodic boundary conditions are assumed: a box with size 85/h Mpc, having cell size 0.31/h Mpc, is followed in a simulation with 270(exp 3) = 10(exp 7.3) cells. We adopt omega = 0.45, lambda = 0.55, h identically equal to H/100 km/s/Mpc = 0.6, and then, from the cosmic background explorer (COBE) and light element nucleosynthesis, sigma(sub 8) = 0.77, omega(sub b) = 0.043. We identify the X-ray emitting clusters in the simulation box, compute the luminosity function at several wavelength bands, the temperature function and estimated sizes, as well as the evolution of these quantities with redshift. This open model succeeds in matching local observations of clusters in contrast to the standard omega = 1, CDM model, which fails. It predicts an order of magnitude decline in the number density of bright (h nu = 2-10 keV) clusters from z = 0 to z = 2 in contrast to a slight increase in the number density for standard omega = 1, CDM model. This COBE-normalized CDM + lambda model produces approximately the same number of X-ray clusters having L(sub x) greater than 10(exp 43) erg/s as observed. The background radiation field at 1 keV due to clusters is approximately the observed background which, after correction for numerical effects, again indicates that the model is consistent with observations.

Cen, Renyue↗

DOE OSTI2022

The short ionizing photon mean free path at z = 6 in Cosmic Dawn III, a new fully coupled radiation-hydrodynamical simulation of the Epoch of Reionization

ABSTRACT Recent determinations of the mean free path of ionizing photons (mfp) in the intergalactic medium (IGM) at z = 6 are lower than many theoretical predictions. In order to gain insight, we investigate the evolution of the mfp in our new massive fully coupled radiation-hydrodynamics cosmological simulation of reionization: Cosmic Dawn III (CoDa III). CoDa III’s scale ($\rm 94^3 \, cMpc^3$) and resolution ($\rm 8192^3$ grid) make it particularly suitable to study the IGM during reionization. The simulation was performed with ramses-cudaton on Summit, and used 13 1072 processors coupled to 24 576 GPUs, making it the largest reionization simulation, and largest ever ramses simulation. A superior agreement with global constraints on reionization is obtained in CoDa III over Cosmic Dawn II (CoDa II), especially for the evolution of the neutral hydrogen fraction and the cosmic photoionization rate, thanks to an improved calibration, later end of reionization (z = 5.6), and higher spatial resolution. Analysing the mfp, we find that CoDa III reproduces the most recent observations very well, from z = 6 to z = 4.6. We show that the distribution of the mfp in CoDa III is bimodal, with short (neutral) and long (ionized) mfp modes, due to the patchiness of reionization and the coexistence of neutral versus ionized regions during reionization. The neutral mode peaks at sub-kpc to kpc scales of mfp, while the ionized mode peak evolves from 0.1 Mpc h−1 at z = 7 to ∼10 Mpc h−1 at z = 5.2. Computing the mfp as the average of the ionized mode provides the best match to the recent observational determinations. The distribution reduces to a single neutral (ionized) mode at z > 13 (z < 5).

Lewis, Joseph S. W. (ORCID:0000000179178474)↗

DOE OSTI2024

Staged Z-pinch radiation-hydrodynamic simulations on a 20-MA driver

We report on one-dimensional simulations of a staged Z-pinch (SZP) fusion scheme using the radiation-hydrodynamics code HYDRA [Marinak et al., Phys. Plasmas 8, 2275 (2001)]. A pulsed-power driver, based on the Z-machine at Sandia National Laboratories, with a peak current around 20 MA, is used to implode a high-Z gas liner surrounding a cylindrical DT fuel target. For the best conditions used in this work, excluding some physical processes, such as mix, we find a “clean” fusion yield of 7.05 MJ (2.5 ×1018 neutrons), thus showing the potential for high yields using this scheme. The design achieving this yield was determined through a study of different liner materials, liner masses, and fuel masses. We also address previous simulations of the SZP setup that used ad hoc limits of the total internal energy of the liner that can artificially increase fusion yields.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1992

Radiative-hydrodynamical simulations of X-ray burst-induced accretion disk coronae

A fully self-consistent radiative hydrodynamical algorithm developed earlier is used to study the 'steady' state coronae in order to simulate the response of the evaporated outflow to a time-dependent irradiation of the disk's surface. It is found that the burst definition is altered at both low and high inclination angles i. At i greater than or approximately equal to 45 deg, the detectable flux is enhanced (as much as 50 percent) by X-rays scattered into the line of sight from the radiation propagation through the corona, and (greater than or approximately equal to 30 percent) by X-rays reflected off the corona when i is less than or approximately equal to 45 deg.

Mella, Fulvio↗

NASA NTRS1992

Three-dimensional hydrodynamical simulations of colliding stars . III - Collisions and tidal captures of unequal-mass main-sequence stars

A 3D smooth-particle hydrodynamic code with 7000 particles is used to simulate collisions between lower main sequence (MS) stars whose masses differ by a factor of five. Collisions between point-mass intruders and MS stars are simulated to study how the physics of the encounters depends on the finite radius of the smaller MS stars. It is found that the two MS stars become gravitationally bound in a physical collision if their relative velocity at infinity is less than a critical velocity Vd. Vd decreases from 1000 km/s in a head-on collision to 150 km/s in a grazing one. If the less massive star is replaced by a point mass, Vd remains the same for grazing collision and tidal encounters but drops to about 600 km/s in head-on collisions. The collision and coalescence of two MS stars resets the nuclear clock of the coalesced star, so it contracts to the MS. Tidal capture of binary stars in globular clusters can only occur in encounters in which the closest approach of the two stars to their center of mass is less than 2.0 times the sum of their radii.

Benz, W.↗

DOE OSTI2022

The development of a high-resolution Eulerian radiation-hydrodynamics simulation capability for laser-driven Hohlraums

Hohlraums are hollow cylindrical cavities with high-Z material walls used to convert laser energy into uniform x-ray radiation drives for inertial confinement fusion capsule implosions and high energy density physics experiments. Credible computational modeling of hohlraums requires detailed modeling and coupling of laser physics, hydrodynamics, radiation transport, heat transport, and atomic physics. We report on improvements to Los Alamos National Laboratory's xRAGE radiation-hydrodynamics code in order to enable hohlraum modeling. xRAGE's Eulerian hydrodynamics and adaptive mesh refinement make it uniquely well suited to study the impacts of multiscale features in hohlraums. In order to provide confidence in this new modeling capability, we demonstrate xRAGE's ability to produce reasonable agreement with data from several benchmark hohlraum experiments. We also use xRAGE to perform integrated simulations of a recent layered high density carbon capsule implosion on the National Ignition Facility in order to evaluate the potential impacts of the capsule support tent, mixed cell conductivity methodologies, plasma transport, and cross-beam energy transfer (XBT). We find that XBT, seeded by plasma flows in the laser entrance hole (LEH), causes a slight decrease in energy coupling to the capsule and that all of these impact the symmetry of the x-ray drive such that they have an appreciable impact on the capsule implosion shape.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

NASA NTRS1998

Three-Dimensional Hydrodynamic Simulations of the Postimpact Proto-Earth

Astrophysical fluid configurations are susceptible to a variety of nonaxisymmetric instabilities under the combined effects of rotation, self-gravity, and thermal pressure. When strong enough, they can induce rapid transport of mass and angular momentum. Our own previous studies of nonaxisymmetric instabilities in model protostars and protostellar disks show that significant transport can occur on orbital timescales and that material can be ejected to large distances. In this contribution, we present three-dimensional simulations of the circumterrestrial debris belt that may have resulted from a giant impact. Our three-dimensional hydrodynamics code with self-gravity and artificial viscosity is fully second-order in space and time; the equations of hydrodynamics and the Poisson equation are solved on an Eulerian cylindrical grid. In the preliminary calculations presented here, we use a simplified EOS where the central proto-Earth is treated as an n = 1/2 polytropic fluid, surrounded by a more compressible, rapidly rotating, fluid disk that represents silicate vapor. Our initial disk parameters are generated from the endstate data of recent smoothed particle hydrodynamics giant-impact calculations. Ultimately, we wish to detennine under what conditions nonaxisymmetric instabilities grow in the postimpact disk and whether they facilitate the transport of material outside the proto-Earth's Roche Limit, leading to the formation of the Moon. In future work, we hope to include a more realistic EOS and the consequences of heating, cooling, and phase transitions.

Pickett, B. K.↗

DOE OSTI2023

Ab-initio General-relativistic Neutrino-radiation Hydrodynamics Simulations of Long-lived Neutron Star Merger Remnants to Neutrino Cooling Timescales

Abstract We perform the first 3D ab-initio general-relativistic neutrino-radiation hydrodynamics of a long-lived neutron star merger remnant spanning a fraction of its cooling timescale. We find that neutrino cooling becomes the dominant energy loss mechanism after the gravitational-wave dominated phase (∼20 ms postmerger). Electron flavor antineutrino luminosity dominates over electron flavor neutrino luminosity at early times, resulting in a secular increase of the electron fraction in the outer layers of the remnant. However, the two luminosities become comparable ∼20–40 ms postmerger. A dense gas of electron antineutrinos is formed in the outer core of the remnant at densities ∼10 14.5 g cm −3 , corresponding to temperature hot spots. The neutrinos account for ∼10% of the lepton number in this region. Despite the negative radial temperature gradient, the radial entropy gradient remains positive, and the remnant is stably stratified according to the Ledoux criterion for convection. A massive accretion disk is formed from the material squeezed out of the collisional interface between the stars. The disk carries a large fraction of the angular momentum of the system, allowing the remnant massive neutron star to settle to a quasi-steady equilibrium within the region of possible, stable, rigidly rotating configurations. The remnant is differentially rotating, but it is stable against the magnetorotational instability. Other MHD mechanisms operating on longer timescales are likely responsible for the removal of the differential rotation. Our results indicate the remnant massive neutron star is thus qualitatively different from a protoneutron stars formed in core-collapse supernovae.

79 ASTRONOMY AND ASTROPHYSICS↗

DOE OSTI2024

Element Formation in Radiation-hydrodynamics Simulations of Kilonovae

Abstract Understanding the details of r -process nucleosynthesis in binary neutron star merger (BNSM) ejecta is key to interpreting kilonova observations and identifying the role of BNSMs in the origin of heavy elements. We present a self-consistent, two-dimensional, ray-by-ray radiation-hydrodynamic evolution of BNSM ejecta with an online nuclear network (NN) up to a timescale of days. For the first time, an initial numerical relativity ejecta profile composed of the dynamical component and spiral-wave and disk winds is evolved including detailed r -process reactions and nuclear heating effects. A simple model for the jet energy deposition is also included. Our simulation highlights that the common approach of relating in postprocessing the final nucleosynthesis yields to the initial thermodynamic profile of the ejecta can lead to inaccurate predictions. Moreover, we find that neglecting the details of the radiation-hydrodynamic evolution of the ejecta in nuclear calculations can introduce deviations of up to 1 order of magnitude in the final abundances of several elements, including very light and second r -process peak elements. The presence of a jet affects element production only in the innermost part of the polar ejecta, and it does not alter the global nucleosynthesis results. Overall, our analysis shows that employing an online NN improves the reliability of nucleosynthesis and kilonova light-curve predictions.

Magistrelli, Fabio (ORCID:0009000509767851)↗

DOE OSTI2024

Hydrodynamic Simulations of Oxygen–Neon Classical Novae as Galactic 7 Li Producers and Potential Accretion-induced Collapse Progenitors*

Abstract We report on studies of classical nova (CN) explosions where we follow the evolution of thermonuclear runaways (TNRs) on oxygen–neon (ONe) white dwarfs (WDs). Using NOVA, a 1D hydrodynamic computer code, we accrete solar matter until the TNR is ongoing and then switch to a mixed composition. This approach is guided by the results of multidimensional studies of TNRs in WDs, which find that sufficient mixing with WD core material occurs after the TNR is well underway, and levels of enrichment of the CNONeMg elements are reached that agree with observations of CN ejecta abundances. Because the amount of accreted material is inversely proportional to the oxygen abundance, by first accreting solar matter, the amount of accreted material is larger than in those simulations with an initially enriched composition. We vary the mass of the WD (from 0.6 M ⊙ to 1.35 M ⊙ ) and the composition of the mixed materials. Our results show large enrichments of 7 Be in the ejected gases, implying that ONe CNe and CO CNe may be responsible for a significant fraction (∼100 M ⊙ ) of the Galactic 7 Li (∼1000 M ⊙ ). The production of 22 Na and 26 Al in CN explosions and the γ -ray emission predicted by our simulations are discussed. The WDs in all our simulations eject less material than they accrete and we predict that the WD is growing in mass as a consequence of the CN outburst. ONe CNe, therefore, may be an important channel for accretion-induced collapse events.

Starrfield, Sumner (ORCID:0000000213596312)↗

DOE OSTI2025

LDRD Abbreviated report: High-Order General-Discrete-Ordinates Method Enabling Efficient Deterministic Transport in Hydrodynamic Simulations

Deterministic transport simulations for national-security and energy applications often operate in high-dimensional phase-space, where accuracy and cost both become major challenges. A common numerical artifact in such problems is the “ray-effect,” which appears as unphysical streaks. Beyond misinterpretation, these artifacts can contaminate tightly coupled physics, such as fluid dynamics, radiation-hydrodynamics, and laser-plasma interactions, eroding the predictive capability of entire multiphysics workflows. Our objective was to make high-dimension studies practical on modern hardware while mitigating the ray-effect without relying on prohibitively expensive sampling approaches such as Monte Carlo methods. We developed the Generic Discretization Library (GenDiL), a Graphics Processing Unit (GPU)-first framework that uses high-order Discontinuous Galerkin (DG) methods and matrix-free algorithms to reduce memory usage and improve computational efficiency, critical for phase-space simulations. GenDiL supports phase-space adaptivity in both mesh size and polynomial order (hp-adaptivity) to place resolution only where it is needed. A central capability is Local Dimensional Refinement (LDR), which couples lower-dimension continuum models to higher-dimension kinetic models through stable and conservative interfaces, so that high-fidelity physics is applied only in regions where it is essential. Building on the GenDiL framework, we developed the General SN (GSN) family of algorithms as a true generalization of the polar SN approach (discrete ordinates, often denoted SN). Rather than tying discrete ordinates to a specific polar change of coordinates, GSN formulates transport on an arbitrary change of coordinates chosen to reduce ray-effect. We studied two complementary variants: an analytic variant, where the coordinate map is prescribed in advance by a closed-form function; and a data-driven variant, where a quantity of interest, such as the net flux, guides the coordinate system. GenDiL provides the library infrastructure for efficient GPU execution, but the GSN concept is algorithmic and independent of any one library. Across representative high-dimension tests, including non-symmetric solutions, both variants delivered strong ray-effect mitigation at practical cost, moving four- to six-dimensional analysis toward repeatable, routine studies.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗

DOE OSTI2024

Achieving Higher Order Accuracy in Space in Hydrodynamic Simulations of Self-Gravitating Gas

Modern astrophysical simulation codes employ a variety of numerical algorithms capable of achieving higher-order accuracy in both space and time. Albeit they succeed in achieving an effective higher spatial resolution and in suppressing the numerical damping of waves, to our knowledge, all current astrophysical simulations invoking self-gravity are limited to second-order accuracy in space. If we can devise an algorithm to evaluate self-gravity with a higher-order spatial accuracy, we can better the evaluation of the gravitational acceleration and gravitational energy release which dictate the evolution of many astrophysical systems. Herein, we present a numerical algorithm for self-gravitating hydrodynamics capable of achieving fourth-order accuracy for a given density distribution on a Cartesian uniform grid. First, we derive the cell-averaged gravitational potential at fourth-order accuracy from the cell-averaged density by solving the Poisson equation. Next, we obtain the cell average of the product of the density and gravitational acceleration, which differs from the cell-averaged density multiplied by the cell-averaged gravitational acceleration. We then show the verification of the algorithm by applying it to critical test problems: (1) maintaining equilibria of self-gravitating slabs, even upon advection, (2) evolving a polytropic sphere with a massive power-law envelope, and (3) conservation of specific entropy during the propagation of a sound wave.

79 ASTRONOMY AND ASTROPHYSICS↗