SPRITZ Simulation Frameworks

• SPRITZ Simulation is a dual-framework approach combining full general-relativistic magnetohydrodynamics for compact objects with semi-empirical extragalactic modeling based on observed data.
• It leverages high-order finite volume methods, robust primitive recovery via RePrimAnd, and innovative divergence control to ensure accurate and stable simulations.
• The extragalactic component uses observational infrared luminosity functions and empirical SED templates to forecast survey outcomes and guide instrument design.

SPRITZ Simulation

The term "SPRITZ Simulation" encompasses several distinct approaches and applications within computational astrophysics, galaxy evolution, engine spray modeling, and information security, all denoted by the use of the acronym "SPRITZ" in different research contexts. Most notably, SPRITZ refers to a series of advanced, open-source simulation frameworks for general relativistic magnetohydrodynamics (GRMHD) in multimessenger astrophysics, and to semi-empirical galaxy simulations designed for infrared extragalactic survey prediction and interpretation. The following account organizes the key methodologies, technical principles, and scientific implications of representative SPRITZ simulations, drawing on the major literature describing both astrophysical and extragalactic classes of the code as well as important applied variations.

1. GRMHD SPRITZ: Structure and Numerical Formulation

The SPRITZ code was developed as a fully general-relativistic magnetohydrodynamics (GRMHD) platform to simulate neutron star mergers and other compact object systems in dynamical spacetimes. It follows the Valencia formulation for conservative evolution of fluid and electromagnetic variables in a 3+1 split of spacetime, employing Cartesian coordinates for three-dimensional simulations (Cipolletta et al., 2019, Cipolletta et al., 2020).

Fluid evolution is governed by equations of the form: $$\frac{1}{\sqrt{-g}} \left[ \partial_t(\sqrt{\gamma} \mathbf{F}^0) + \partial_i (\sqrt{-g} \mathbf{F}^i) \right] = \mathbf{S}$$ where $\mathbf{F}^0$ is the vector of conserved variables (rest-mass density $D = \rho W$, momentum, energy, and magnetic components), and $\mathbf{S}$ represents source terms. The spatial metric $\gamma_{ij}$, lapse function $\alpha$, and shift vector $\beta^i$ define the spacetime line element: $$ds^2 = -(\alpha^2-\beta_i\beta^i)dt^2 + 2\beta_i dx^i dt + \gamma_{ij} dx^i dx^j$$ The electromagnetic sector is advanced using a vector potential formalism (staggered or non-staggered), with magnetic fields constructed via $B^i = \epsilon^{ijk} \partial_j A_k$, automatically enforcing the divergence-free constraint $\nabla\cdot\mathbf{B} = 0$.

Spacetime dynamics are treated via the BSSNOK formalism, coupling Einstein's equations with the GRMHD evolution.

2. Equation of State, Microphysics, and Neutrino Leakage

SPRITZ incorporates a modular equation of state (EOS) driver, EOS_Omni, enabling the use of tabulated, finite-temperature EOSs or analytic gamma-law/polytropic forms. For example: $$p_{\rm gas} = (\Gamma - 1)\rho\epsilon \quad \text{or} \quad p_{\rm gas} = K\rho^\Gamma$$ This setup allows flexible simulations covering cold to hot, composition-dependent nuclear matter relevant for neutron star interiors and merger ejecta (Cipolletta et al., 2019).

Advanced versions of SPRITZ (Cipolletta et al., 2020) implement an approximate neutrino leakage scheme derived from ZelmaniLeak, computing both energy and lepton number loss and gain due to neutrino emission and reabsorption. Local emission rates are interpolated between "free" and "diffusive" regimes based on optical depth, for each neutrino species (ν_e, $\bar{\nu}_e$, ν_x). The effect of neutrino irradiation (heating) is included via local absorption, crucial for post-merger remnant evolution and jet launching models.

3. Numerical Algorithm Innovations and Recovery Schemes

SPRITZ utilizes high-order finite volume methods: fifth-order Weighted Essentially Non-Oscillatory (WENOZ) reconstructions and, where required, sixth-order accurate flux derivative corrections. All evolved variables, including the magnetic field in divergence-cleaning implementations, are cell-centered, simplifying mesh refinement and facilitating uniform high-order treatment (Neuweiler et al., 30 Jul 2024).

A critical algorithmic advance is the integration of RePrimAnd for the conservative-to-primitive variable recovery problem (Kalinani et al., 2021). This algebraic root-finding approach ensures unique, robust, and analytically bounded recovery of primitive variables even in regimes with extreme magnetization or low density, a previously limiting factor in the accuracy and stability of GRMHD codes beyond previous Noble-type or multi-dimensional iterative schemes: $\mu = (W h)^{-1}$ uniquely determines the full primitive state, accommodating rigorous error bounds and corrective policies if the state falls outside the physically valid manifold.

4. Code Validation and Physical Applications

A comprehensive battery of tests validates the GRMHD SPRITZ implementation:

• In special and general relativistic regimes, the code accurately reproduces standard shock tube (Balsara) tests, cylindrical and spherical explosions, magnetic rotors, Alfvén wave convergence (showing $\propto (\Delta x)^2$ scaling), and Tolman-Oppenheimer-Volkoff (TOV) neutron star models, confirming correct primitive recovery and dynamical evolution (Cipolletta et al., 2019, Cipolletta et al., 2020, Kalinani et al., 2021).
• Mesh refinement is shown to be robust under cell-centered divergence cleaning.
• In challenging scenarios (e.g., collapse of hypermassive neutron stars or Fishbone–Moncrief black hole-accretion disks), RePrimAnd achieves convergence in regions with magnetic-to-fluid pressure ratios $\beta^{-1}$ up to $10^4$, where prior schemes fail (Kalinani et al., 2021).
• Full BNS mergers with neutrino leakage, magnetic field amplification, and shearing instabilities display physical agreement and convergence with other state-of-the-art codes (Neuweiler et al., 30 Jul 2024), with differences largely set by reconstruction order and flux conservation at AMR boundaries.

5. Comparative Features: Divergence Control and Algorithmic Strategies

SPRITZ has evolved substantially in its strategy for enforcing $\nabla\cdot\mathbf{B} = 0$:

• Original versions implemented a staggered vector potential, yielding superior suppression of post-shock oscillations compared to non-staggered approaches and naturally enforcing the solenoidal constraint by construction (Cipolletta et al., 2019).
• In later high-order, cell-centered implementations motivated by ease of AMR and reconstructor homogeneity, hyperbolic divergence cleaning is adopted (Neuweiler et al., 30 Jul 2024). Here, a damping-cleaning field ($\zeta$) is evolved concurrently,

$$\partial_t \hat{B}^j + \partial_i \left[(\alpha v^i - \beta^i)\hat{B}^j - \alpha v^j \hat{B}^i + \alpha \sqrt{\gamma}\gamma^{ij}\zeta \right] = -\hat{B}^i \partial_i \beta^j + \zeta \partial_i(\alpha \sqrt{\gamma}\gamma^{ij})$$

yielding residual divergences on the order of $10^{-3}$ (normalized), subdominant to fluid discretization errors, but less precise than flux-corrected finite-volume schemes.

Comparison with BAM, SACRA$_{\rm KK22}$, and other platforms indicates none of these divergence preservation options dominate the accuracy budget in typical BNS merger scenarios; primary sensitivity arises from flux reconstruction method, primitive recovery robustness, and mesh boundary treatment (Neuweiler et al., 30 Jul 2024).

6. Semi-Empirical and Extragalactic SPRITZ: Phenomenological Simulations

Distinct from the GRMHD code, the "SPRITZ" simulation suite ("Spectro-Photometric Realisations of Infrared-selected Targets at all-z") denotes a semi-empirical framework for extragalactic sky simulations, used for planning and interpreting deep infrared surveys (Bisigello et al., 2020, Bisigello et al., 2021, Bisigello et al., 2022, Traina et al., 1 Sep 2025, Cunha et al., 5 Mar 2025).

• Construction: Populations are sampled from observed infrared luminosity functions (LFs) and galaxy stellar mass functions (GSMFs), not from cosmological N-body merger trees. Empirical and semi-empirical spectral energy distribution (SED) templates are assigned to each, with physical properties (stellar mass, star-formation rate, AGN contribution, metallicity) attached via observed scaling relations.
• Applications: The simulation produces validated number counts, kernel density distributions in SFR–$M_*$ space, and luminosity functions for mock galaxies, AGN, and composite systems, matching observed multi-wavelength constraints from UV to radio.
• Line emission: [CII] and CO line luminosities ($J\leq24$) are derived from SFR or L$_{\rm IR}$ using various empirical relations, with metallicity-dependent conversions preferred for [CII]: $$\log(L_{\rm [CII]}/L_\odot) = 7.0 + 1.2\log(\rm SFR) + 0.021\log(Z/Z_\odot) + 0.012\log(\rm SFR)\log(Z/Z_\odot) - 0.74[\log(Z/Z_\odot)]^2$$ Yielding molecular gas mass functions and line luminosity functions in close agreement with current data if a [CII]-to-H$_2$ conversion factor near 130 $M_\odot/L_\odot$ is adopted (Bisigello et al., 2022).
• Survey prediction: SPRITZ is utilized to forecast outcomes of infrared missions (e.g., SPICA, PRIMA), providing mock images, photometric catalogues, and simulated spectroscopy. The simulations underpin planning for PRIMA’s dust mass function recovery and the paper of obscured AGN populations (Traina et al., 1 Sep 2025, Cunha et al., 5 Mar 2025).

7. Impact, Limitations, and Future Prospects

SPRITZ and its associated simulation frameworks deliver flexible, high-precision modeling tools in both computational astrophysics (GRMHD) and extragalactic survey design. The main strengths are:

• For GRMHD: Unified primitive recovery, cell-centered high-order accuracy, robust solenoidal constraint strategies, and flexible EOS/microphysics modules ready for full microphysical merger simulations.
• For extragalactic surveys: Empirically validated mock catalogues covering a large dynamic range in galaxy properties, validated predictions for emission-line diagnostics, and forecasted survey outcomes for infrared facilities.

Limitations reside in residual divergence errors (for cell-centered divergence cleaning), intrinsic scatter in empirical star-formation–to–line-luminosity calibrations, and reliance on current observational calibrations, especially for high-redshift and faint-end populations.

Ongoing and future developments include full coupling with advanced neutrino transport (e.g., Monte Carlo), enhancements to the EOS and feedback modules, public code releases, and continued integration with next-generation survey data for both simulation branches.

In summary, the SPRITZ simulation frameworks, encompassing both GRMHD and extragalactic phenomenological modeling, constitute a set of advanced, publicly accessible numerical tools that have become central to quantitative interpretation and planning in multimessenger astrophysics and extragalactic survey science (Cipolletta et al., 2019, Bisigello et al., 2020, Cipolletta et al., 2020, Kalinani et al., 2021, Bisigello et al., 2022, Neuweiler et al., 30 Jul 2024, Bisigello et al., 2021, Traina et al., 1 Sep 2025, Cunha et al., 5 Mar 2025).