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In this paper, we propose and study first- and second-order (in time) stabilized linear finite element schemes for the incompressible Navier-Stokes (NS) equations. The energy, momentum, and angular momentum conserving (EMAC) formulation has emerged as a promising approach for conserving energy, momentum, and angular momentum of the NS equations, while the exponential scalar auxiliary variable (ESAV) has become a popular technique for designing linear energy-stable numerical schemes. Our method leverages the EMAC formulation and the Taylor-Hood element with grad-div stabilization for spatial discretization. We adopt the implicit-explicit backward differential formulas (BDFs) coupled with a novel stabilized ESAV approach for time stepping. For the solution process, we develop an efficient decoupling technique for the resulting fully-discrete systems so that only one linear Stokes solve is needed at each time step, which is similar to the cost of classic implicit-explicit BDF schemes for the NS equations. Robust optimal error estimates are successfully derived for both velocity and pressure for the two proposed schemes, with Gronwall constants that are particularly independent of the viscosity. Furthermore, it is rigorously shown that the grad-div stabilization term can greatly alleviate the viscosity-dependence of the mesh size constraint, which is required for error estimation when such a term is not present in the schemes. Various numerical experiments are conducted to verify the theoretical results and demonstrate the effectiveness and efficiency of the grad-div and ESAV stabilization strategies and their combination in the proposed numerical schemes, especially for problems with high Reynolds numbers. 

Abstract We present ∼8–40μm SOFIA-FORCAST images of seven regions of “clustered” star formation as part of the SOFIA Massive Star Formation Survey. We identify a total of 34 protostar candidates and build their spectral energy distributions (SEDs). We fit these SEDs with a grid of radiative transfer models based on the turbulent core accretion (TCA) theory to derive key protostellar properties, including initial core mass,M_c, clump environment mass surface density, Σ_cl, and current protostellar mass,m_*. We also carry out empirical graybody (GB) estimation of Σ_cl, which allows a case of restricted SED fitting within the TCA model grid. We also release version 2.0 of the open-source Python packagesedcreator, which is designed to automate the aperture photometry and SED building and fitting process for sources in clustered environments, where flux contamination from close neighbors typically complicates the process. Using these updated methods, SED fitting yields values ofM_c∼ 30–200M_⊙, Σ_cl,SED∼ 0.1–3 g cm⁻², andm_*∼ 4–50M_⊙. The GB fitting yields smaller values of Σ_cl,GB≲ 1 g cm⁻². From these results, we do not find evidence for a critical Σ_clneeded to form massive (≳8M_⊙) stars. However, we do find tentative evidence for a dearth of the most massive (m_*≳ 30M_⊙) protostars in the clustered regions, suggesting a potential impact of environment on the stellar initial mass function. 

Abstract We present ∼10–40μm SOFIA-FORCAST images of 11isolatedprotostars as part of the SOFIA Massive (SOMA) Star Formation Survey, with this morphological classification based on 37μm imaging. We develop an automated method to define source aperture size using the gradient of its background-subtracted enclosed flux and apply this to build spectral energy distributions (SEDs). We fit the SEDs with radiative transfer models, developed within the framework of turbulent core accretion (TCA) theory, to estimate key protostellar properties. Here, we release the sedcreator python package that carries out these methods. The SEDs are generally well fitted by the TCA models, from which we infer initial core massesM_cranging from 20–430M_⊙, clump mass surface densities Σ_cl∼ 0.3–1.7 g cm⁻², and current protostellar massesm_*∼ 3–50M_⊙. From a uniform analysis of the 40 sources in the full SOMA survey to date, we find that massive protostars form across a wide range of clump mass surface density environments, placing constraints on theories that predict a minimum threshold Σ_clfor massive star formation. However, the upper end of them_*−Σ_cldistribution follows trends predicted by models of internal protostellar feedback that find greater star formation efficiency in higher Σ_clconditions. We also investigate protostellar far-IR variability by comparison with IRAS data, finding no significant variation over an ∼40 yr baseline.