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1. Entropy-Conserving Scheme for Modeling Nonthermal Energies in Fluid Dynamics Simulations

   Semenov, Vadim A.; Kravtsov, Andrey V; Diemer, Benedikt (August 2021, The astrophysical journal)

   null (Ed.)

   We compare the performance of energy-based and entropy-conservative schemes for modeling nonthermal energy components, such as unresolved turbulence and cosmic rays, using idealized fluid dynamics tests and isolated galaxy simulations. While both methods are aimed to model advection and adiabatic compression or expansion of different energy components, the energy-based scheme numerically solves the non-conservative equation for the energy density evolution, while the entropy-conserving scheme uses a conservative equation for modified entropy. Using the standard shock tube and Zel'dovich pancake tests, we show that the energy-based scheme results in a spurious generation of nonthermal energy on shocks, while the entropy-conserving method evolves the energy adiabatically to machine precision. We also show that, in simulations of an isolated Lstar galaxy, switching between the schemes results in ~20-30% changes of the total star formation rate and a significant difference in morphology, particularly near the galaxy center. We also outline and test a simple method that can be used in conjunction with the entropy-conserving scheme to model the injection of nonthermal energies on shocks. Finally, we discuss how the entropy-conserving scheme can be used to capture the kinetic energy dissipated by numerical viscosity into the subgrid turbulent energy implicitly, without explicit source terms that require calibration and can be rather uncertain. Our results indicate that the entropy-conserving scheme is the preferred choice for modeling nonthermal energy components, a conclusion that is equally relevant for Eulerian and moving-mesh fluid dynamics codes. 

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2. The Importance of Heat Flux in Low-Mach-number, Quasi-parallel Collisionless Shocks

   https://doi.org/10.3847/1538-4357/ae2acc  

   Haggerty, Colby C; Caprioli, Damiano; Cassak, Paul A; Barbhuiya, M Hasan; Wilson, Lynn; Turner, Drew L (February 2026, The Astrophysical Journal)

   Abstract Collisionless plasma shocks are a common feature of many space and astrophysical systems. They are sources of high-energy particles and nonthermal emission, channeling as much as 20% of the shock’s energy into nonthermal particles. The generation and acceleration of these nonthermal particles have been previously studied and shown to affect shock hydrodynamics to the zeroth order. In this work, we use self-consistent hybrid particle-in-cell simulations to examine the effect of self-generated nonthermal ion populations on the nature of collisionless, quasi-parallel shocks. Accelerated nonthermal particles downstream of the shock diffuse into the upstream region, taking energy away from the shock, which increases the compression ratio, slows the shock down, and flattens the nonthermal population’s spectral index for lower-Mach-number shocks. We show that this enhances shock compressibility when the heat flux is included in the Rankine–Hugoniot jump conditions, results that are roughly consistent with previous theories of “cosmic-ray-modified shocks.” Additionally, the simulation data show that heat flux and enthalpy flux cancels out in the upstream region, yielding a relatively simple, alternative closure for the jump conditions which accurately predict for the shock speed and compression ratio. The results have the potential to explain discrepancies between predictions and observations in a wide range of systems, such as inaccuracies in predictions of the arrival times of coronal mass ejections and the conflicting radio and X-ray observations of intracluster shocks. These effects will likely need to be included in fluid modeling to predict shock evolution accurately. 

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3. Energy Partition at a Collisionless Supercritical Quasi‐Parallel Shock

   https://doi.org/10.1029/2025JA034718  

   Schwartz, Steven J; Trattner, Karlheinz J; Raptis, Savvas; Ergun, Robert E; Wilson, Lynn B; Gomez, Roman G; Cohen, Ian J; Gershman, Daniel J; Goodrich, Katherine A; Kucharek, Harald; et al (March 2026, Journal of Geophysical Research: Space Physics)

   Abstract Shocks in collisionless astrophysical plasmas redistribute some of the incident flow energy into both thermal and non‐thermal energy. Quantifying the partition of that energy amongst various particle species or their sub‐populations, and electromagnetic energy, represents a fundamental goal of shock physics. It embodies the role of the equation of state for the system. Here we apply a framework to assess all the incident and downstream energy fluxes at a crossing of Earth's bow shock for which the upstream magnetic field was roughly aligned with the shock normal direction. Such quasi‐parallel shocks are known to be non‐steady and to produce significant populations of suprathermal particles. We quantify the evolution of all the important carriers of energy flux through the shock region. We sub‐divide the proton population into thermal, suprathermal, and energetic components in order to investigate the shock's efficiency in energizing the nonthermal particles. While the largest energy fluxes are found in the incident proton ram energy and downstream proton thermal enthalpy fluxes, a significant suprathermal population pervades the regions both up‐ and downstream. We also evaluate the energy fluxes attributable to fluctuations in the fluid and field parameters. 

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4. Electron Injection via Modified Diffusive Shock Acceleration in High-Mach-number Collisionless Shocks

   https://doi.org/10.3847/2041-8213/ad0cf9  

   Grassi, A.; Rinderknecht, H. G.; Swadling, G. F.; Higginson, D. P.; Park, H.-S.; Spitkovsky, A.; Fiuza, F. (November 2023, The Astrophysical Journal Letters)

   Abstract The ability of collisionless shocks to efficiently accelerate nonthermal electrons via diffusive shock acceleration (DSA) is thought to require an injection mechanism capable of preaccelerating electrons to high enough energy where they can start crossing the shock front potential. We propose, and show via fully kinetic plasma simulations, that in high-Mach-number shocks electrons can be effectively injected by scattering in kinetic-scale magnetic turbulence produced near the shock transition by the ion Weibel, or current filamentation, instability. We describe this process as a modified DSA mechanism where initially thermal electrons experience the flow velocity gradient in the shock transition and are accelerated via a first-order Fermi process as they scatter back and forth. The electron energization rate, diffusion coefficient, and acceleration time obtained in the model are consistent with particle-in-cell simulations and with the results of recent laboratory experiments where nonthermal electron acceleration was observed. This injection model represents a natural extension of DSA and could account for electron injection in high-Mach-number astrophysical shocks, such as those associated with young supernova remnants and accretion shocks in galaxy clusters. 

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5. A mass- and energy-conserved DG method for the Schrödinger-Poisson equation

   https://doi.org/10.1007/s11075-021-01139-0  

   Yi, Nianyu; Liu, Hailiang (February 2021, Numerical Algorithms)

   null (Ed.)

   In this paper, we construct, analyze, and numerically validate conservative discontinuous Galerkin (DG) schemes for approximating the Schr\"{o}dinger-Poisson equation. The proposed schemes all satisfy both mass and energy conservation. For the semi-discrete DG scheme optimal $L^2$ error estimates are obtained. Efficient iterative solvers are also constructed to solve the second order implicit time discretization. A number of numerical tests are presented to demonstrate the method’s accuracy and robustness, confirming that both mass and energy are well preserved over long time simulations. 

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