KORAL: GRRMHD Code & SSD Analysis

• KORAL is a dual-purpose system that encompasses a general relativistic radiation hydrodynamics and magnetohydrodynamics code for simulating black-hole accretion, as well as a knowledge-graph framework for SSD analysis.
• The astrophysical KORAL employs conservative finite-volume schemes with a covariant M1 closure, advanced electron-heating models, and two-temperature plasma evolution to capture complex accretion phenomena.
• The SSD KORAL integrates data and literature knowledge graphs with large language models, achieving high precision in full-spectrum analysis and offering actionable SSD operational insights.

KORAL denotes two distinct research systems in the cited literature. In relativistic astrophysics, KORAL is a conservative code for general relativistic radiation hydrodynamics and magnetohydrodynamics that has been used for super-critical accretion, low-luminosity black-hole flows, two-temperature plasmas, nonthermal electrons, tidal disruption events, horizon-scale imaging, and hybrid GRMHD–GRFFE jet simulations (Sadowski et al., 2012). In storage-systems research, “KORAL” also denotes a knowledge-driven reasoning framework that combines a Data Knowledge Graph, a Literature Knowledge Graph, and a LLM for full-spectrum SSD operational analysis (Akewar et al., 10 Feb 2026).

1. Historical development and research domains

The astrophysical KORAL first appeared as a multi-dimensional, conservative general relativistic radiation hydrodynamics code with a covariant M1 closure and semi-implicit treatment of radiative source terms, explicitly designed to handle optically thin, intermediate, and optically thick regimes in curved spacetime (Sadowski et al., 2012). It was then extended to general relativistic radiation magnetohydrodynamics and used for the first axisymmetric simulations of super-critical magnetized accretion in general relativity, with applications to Schwarzschild and Kerr black holes accreting at $\dot M \sim 100\text{--}200\,\dot M_{\rm Edd}$ (Sadowski et al., 2013).

Subsequent papers turned KORAL into a broader GRRMHD platform. The cited literature describes additions including a sub-grid magnetic dynamo for long-duration axisymmetric radiative disks, thermal Comptonization, radiative viscosity for M1, photon-conserving Comptonization, relativistic two-temperature thermodynamics, nonthermal electron evolution, electron-heating closures calibrated to plasma microphysics, and a hybrid GRMHD+GRFFE method for highly magnetized funnels (Sadowski et al., 2014). The same code base has also been run in stripped-down modes: pure ideal-GRMHD with radiation and two-temperature physics disabled for code-comparison work, and special-relativistic non-resistive MHD for Orszag–Tang reconnection benchmarks (Porth et al., 2019).

A separate 2026 usage adopts the same name for a storage-systems framework: “KORAL: Knowledge Graph Guided LLM Reasoning for SSD Operational Analysis” (Akewar et al., 10 Feb 2026). This second KORAL is unrelated to the astrophysical code except for the shared acronym.

2. Core relativistic formulation of the astrophysical code

In its astrophysical usage, KORAL evolves fluid and radiation in a fixed spacetime metric, usually in horizon-penetrating Kerr–Schild or modified Kerr–Schild coordinates. The basic conservation laws are

$\nabla_\mu(\rho u^\mu)=0,$

$$\nabla_\mu\!\left(T^{\mu\nu}_{\rm MHD}+R^{\mu\nu}\right)=0,$$

together with the ideal-MHD induction equation advanced in conservative form and the discrete maintenance of $\nabla\!\cdot\!B=0$ by constrained transport (Chael et al., 2018). For ideal GRMHD, the matter-plus-field stress–energy tensor is written as

$$T^{\mu\nu}_{\rm MHD}=(\rho+u+p+b^2)\,u^\mu u^\nu + \left(p+\frac{b^2}{2}\right)g^{\mu\nu}-b^\mu b^\nu,$$

or in closely related notation with $u_{\rm g}$ and $p_{\rm g}$ for gas internal energy and pressure (Sadowski et al., 2014).

Radiation is represented by moment equations closed with the covariant M1 closure. In the formulation used across the cited papers, KORAL reconstructs the radiation tensor from a radiation rest-frame energy density and a timelike radiation four-velocity,

$$R^{\mu\nu}=\frac{4}{3}E_R u_R^\mu u_R^\nu+\frac{1}{3}E_R g^{\mu\nu},$$

which yields the correct diffusion and free-streaming limits but inherits the standard M1 limitation for strongly multi-directional radiation fields, such as crossing beams (Sadowski et al., 2012). Gas–radiation coupling enters through a covariant four-force $G^\mu$, with absorption, scattering, and, in later versions, Compton energy exchange.

Numerically, KORAL is described as a conservative Godunov finite-difference or finite-volume code. The recurring algorithmic pattern is explicit treatment of transport terms and semi-implicit treatment of stiff local source terms. In commonly used GRMHD configurations, the code uses piecewise parabolic reconstruction, local Lax–Friedrichs fluxes, second-order time integration, and Flux-CT constrained transport; in its original GRRHD presentation, the same framework was combined with local Newton iterations for the semi-implicit radiation solve (Porth et al., 2019).

3. Thermodynamics, radiation physics, and later extensions

A major extension was the introduction of relativistic two-temperature plasma evolution. In that formulation, electrons and ions share the same bulk four-velocity but carry separate entropies and temperatures, with source terms for viscous heating, Coulomb coupling, and radiative cooling: $$T_{\rm e}(n s_{\rm e}u^\mu)_{;\mu}=\delta_{\rm e}q^{\rm v}+q^{\rm C}-\hat G^0, \qquad T_{\rm i}(n s_{\rm i}u^\mu)_{;\mu}=(1-\delta_{\rm e})q^{\rm v}-q^{\rm C}.$$ KORAL identifies the total dissipative heating $\nabla_\mu(\rho u^\mu)=0,$0 by comparing the evolved total internal energy with the energies implied by adiabatically evolved species entropies, and then partitions this dissipation using a sub-grid electron-heating prescription (Sadowski et al., 2016).

The electron-heating closure became a major scientific lever. Early two-temperature work used the strongly $\nabla_\mu(\rho u^\mu)=0,$1-dependent Landau-damped turbulent cascade model of Howes (2010). Later Sgr A* simulations contrasted that closure with a fit to trans-relativistic anti-parallel magnetic reconnection from particle-in-cell simulations by Rowan et al. (2017). In those runs, the turbulent model heated electrons preferentially in low-$\nabla_\mu(\rho u^\mu)=0,$2 polar regions, while the reconnection model yielded a more uniform $\nabla_\mu(\rho u^\mu)=0,$3 that never exceeded $\nabla_\mu(\rho u^\mu)=0,$4 (Chael et al., 2018).

Radiative microphysics was progressively refined. Thermal Comptonization and a calibrated radiative-viscosity term were added to stabilize M1 transport near the axis in radiative disk simulations (Sadowski et al., 2014). Photon-conserving Comptonization then replaced the assumption that radiation is locally a perfect blackbody by evolving photon number density and treating the radiation field as a Bose–Einstein fluid; the cited simulations found that blackbody Comptonization underestimated gas and radiation temperatures by up to a factor of two and allowed color-correction factors as large as $\nabla_\mu(\rho u^\mu)=0,$5 in the funnel region (Sadowski et al., 2015).

KORAL was also extended beyond thermal electrons. A nonthermal module evolves an isotropic electron distribution $\nabla_\mu(\rho u^\mu)=0,$6 in Lorentz-factor space in parallel with thermal ions, thermal electrons, and radiation, with advection, adiabatic gains and losses, Coulomb coupling, synchrotron, bremsstrahlung, inverse Compton cooling, and a prescribed power-law injection tied to the local electron heating rate (Chael et al., 2017). More recently, a hybrid GRMHD+GRFFE scheme was implemented for regions with extreme magnetization. Above a transition $\nabla_\mu(\rho u^\mu)=0,$7, KORAL switches to force-free electrodynamics while passively evolving field-parallel velocity, plasma density, and plasma energy density, allowing jet-funnel magnetizations as large as $\nabla_\mu(\rho u^\mu)=0,$8 without mass floors (Chael, 2024).

4. Scientific applications in black-hole and transient astrophysics

KORAL’s early GRRMHD applications established the gross phenomenology of super-critical accretion. In axisymmetric simulations of magnetized flows accreting at $\nabla_\mu(\rho u^\mu)=0,$9, both the non-spinning and spinning black-hole models produced optically and geometrically thick disks with funnels through which relativistic gas, Poynting flux, and radiative flux escaped. The net energy outflow efficiencies were $$\nabla_\mu\!\left(T^{\mu\nu}_{\rm MHD}+R^{\mu\nu}\right)=0,$$0 for $$\nabla_\mu\!\left(T^{\mu\nu}_{\rm MHD}+R^{\mu\nu}\right)=0,$$1 and $$\nabla_\mu\!\left(T^{\mu\nu}_{\rm MHD}+R^{\mu\nu}\right)=0,$$2 for $$\nabla_\mu\!\left(T^{\mu\nu}_{\rm MHD}+R^{\mu\nu}\right)=0,$$3, while the radiative luminosities were only $$\nabla_\mu\!\left(T^{\mu\nu}_{\rm MHD}+R^{\mu\nu}\right)=0,$$4, implying radiative efficiencies of order $$\nabla_\mu\!\left(T^{\mu\nu}_{\rm MHD}+R^{\mu\nu}\right)=0,$$5 and strong photon trapping (Sadowski et al., 2013).

The 2D sub-grid dynamo study extended that picture to long-duration radiative disk evolution. With the dynamo active, KORAL sustained turbulence in axisymmetry and recovered 3D-like magnetic statistics, including $$\nabla_\mu\!\left(T^{\mu\nu}_{\rm MHD}+R^{\mu\nu}\right)=0,$$6, $$\nabla_\mu\!\left(T^{\mu\nu}_{\rm MHD}+R^{\mu\nu}\right)=0,$$7, and $$\nabla_\mu\!\left(T^{\mu\nu}_{\rm MHD}+R^{\mu\nu}\right)=0,$$8 outside the ISCO. For non-spinning super-Eddington disks, the total efficiency at infinity remained nearly constant at $$\nabla_\mu\!\left(T^{\mu\nu}_{\rm MHD}+R^{\mu\nu}\right)=0,$$9 over $\nabla\!\cdot\!B=0$0 and $\nabla\!\cdot\!B=0$1, while spinning models reached $\nabla\!\cdot\!B=0$2, $\nabla\!\cdot\!B=0$3, and $\nabla\!\cdot\!B=0$4, with $\nabla\!\cdot\!B=0$5 behavior characteristic of Blandford–Znajek-like extraction. The same study also found runaway cooling and disk collapse for $\nabla\!\cdot\!B=0$6, consistent with radiation-pressure thermal instability (Sadowski et al., 2014).

At low accretion rates, KORAL has been used to study horizon-scale sources. Two-temperature radiative simulations of low-luminosity accretion showed that radiation is dynamically negligible at $\nabla\!\cdot\!B=0$7 but materially alters the flow by $\nabla\!\cdot\!B=0$8, where the disk becomes cooler and geometrically less thick (Sadowski et al., 2016). In Sgr A* simulations with separate electron and ion thermodynamics, KORAL produced 230 GHz images with distinct black hole shadows consistent with the Event Horizon Telescope size measurement, but none of the thermal models reproduced the observed radio slope, the large near-infrared and X-ray flares, or the near-infrared spectral index; the paper therefore argued that non-thermal electrons are required for a full explanation of the emission (Chael et al., 2018). In a radiative MAD model for M87*, KORAL predicted that electron cooling lowers inner-disk $\nabla\!\cdot\!B=0$9 by more than an order of magnitude relative to non-radiative models and narrows the hot jet sheath, with direct implications for ngEHT morphology and variability studies (Chatterjee et al., 2022).

The code has also been used outside steady accretion problems. In tidal disruption event simulations, a novel interior stream-injection boundary allowed KORAL to capture the returning debris stream, disk formation, self-intersection shocks, and emergent radiation simultaneously. For $$T^{\mu\nu}_{\rm MHD}=(\rho+u+p+b^2)\,u^\mu u^\nu + \left(p+\frac{b^2}{2}\right)g^{\mu\nu}-b^\mu b^\nu,$$0 black holes, the resulting disks remained only mildly circularized over the simulated few days, emitted $$T^{\mu\nu}_{\rm MHD}=(\rho+u+p+b^2)\,u^\mu u^\nu + \left(p+\frac{b^2}{2}\right)g^{\mu\nu}-b^\mu b^\nu,$$1, and had radiative efficiencies $$T^{\mu\nu}_{\rm MHD}=(\rho+u+p+b^2)\,u^\mu u^\nu + \left(p+\frac{b^2}{2}\right)g^{\mu\nu}-b^\mu b^\nu,$$2, with highly asymmetric photospheres and possible soft X-ray visibility for favorable viewing angles (Curd, 2021). In a different axisymmetric optically thin setting, KORAL was modified to include the Cosmic Battery term in the induction equation; with that term active, a SANE flow evolved toward a MAD state, accumulating horizon-threading magnetic flux that was absent in the control run (Contopoulos et al., 2017).

5. Verification, inter-code comparisons, and numerical limitations

The original radiation module was validated against radiative shock tubes, a static radiation-pressure-supported atmosphere, shadows, beams of light in curved spacetime, and radiative Bondi accretion, explicitly demonstrating the advantages of M1 over Eddington closure and flux-limited diffusion in problems that require both diffusion and free streaming (Sadowski et al., 2012). The early GRRMHD version was further checked against radiation-modified MHD linear waves, magnetic shock tubes, and the Orszag–Tang vortex (Sadowski et al., 2013).

KORAL has also been benchmarked directly against other production codes. In the Event Horizon GRMHD Code Comparison Project, it was run as a pure ideal-GRMHD solver on a turbulent SANE accretion flow around a Kerr black hole with $$T^{\mu\nu}_{\rm MHD}=(\rho+u+p+b^2)\,u^\mu u^\nu + \left(p+\frac{b^2}{2}\right)g^{\mu\nu}-b^\mu b^\nu,$$3. At $$T^{\mu\nu}_{\rm MHD}=(\rho+u+p+b^2)\,u^\mu u^\nu + \left(p+\frac{b^2}{2}\right)g^{\mu\nu}-b^\mu b^\nu,$$4 resolution, KORAL yielded $$T^{\mu\nu}_{\rm MHD}=(\rho+u+p+b^2)\,u^\mu u^\nu + \left(p+\frac{b^2}{2}\right)g^{\mu\nu}-b^\mu b^\nu,$$5, $$T^{\mu\nu}_{\rm MHD}=(\rho+u+p+b^2)\,u^\mu u^\nu + \left(p+\frac{b^2}{2}\right)g^{\mu\nu}-b^\mu b^\nu,$$6, $$T^{\mu\nu}_{\rm MHD}=(\rho+u+p+b^2)\,u^\mu u^\nu + \left(p+\frac{b^2}{2}\right)g^{\mu\nu}-b^\mu b^\nu,$$7, and $$T^{\mu\nu}_{\rm MHD}=(\rho+u+p+b^2)\,u^\mu u^\nu + \left(p+\frac{b^2}{2}\right)g^{\mu\nu}-b^\mu b^\nu,$$8, placing it within the mature multi-code cluster for near-horizon SANE problems (Porth et al., 2019). In a later Orszag–Tang benchmark against PLUTO, KORAL was used in non-resistive special-relativistic and non-relativistic MHD. At sufficient resolution it reproduced fast, plasmoid-mediated reconnection with $$T^{\mu\nu}_{\rm MHD}=(\rho+u+p+b^2)\,u^\mu u^\nu + \left(p+\frac{b^2}{2}\right)g^{\mu\nu}-b^\mu b^\nu,$$9 in 2D Ideal-MHD, $u_{\rm g}$0 in 2D Rel-MHD, and $u_{\rm g}$1 in 3D, while retaining slightly more magnetic energy and resolving more substructures than PLUTO, albeit with larger total-energy residuals in the relativistic case ($u_{\rm g}$2 at $u_{\rm g}$3, versus $u_{\rm g}$4 for PLUTO) (Kayanikhoo et al., 2023).

The limitations emphasized in the cited literature are correspondingly specific. M1 cannot represent complex angular distributions such as genuinely crossing beams and may create axial artifacts unless regularized (Sadowski et al., 2012). Axisymmetric runs with a sub-grid dynamo remain approximations to 3D turbulence and cannot faithfully capture non-axisymmetric interchange instabilities or sustained MAD states (Sadowski et al., 2014). In low-density polar funnels, temperatures and emissivities can become floor-dominated, which is why several Sgr A* post-processing pipelines exclude the innermost polar zones (Chael et al., 2018). Standard GRMHD also becomes fragile above $u_{\rm g}$5, a difficulty that motivated the hybrid GRMHD+GRFFE extension (Chael, 2024). A common misconception is therefore that KORAL is a single fixed solver; the cited record instead shows a modular framework whose physics content ranges from pure ideal-GRMHD to full two-temperature radiative and nonthermal models.

6. KORAL as a knowledge-graph framework for SSD operations

A distinct 2026 paper uses the name KORAL for a knowledge-driven reasoning framework for SSD operational analysis. This KORAL combines a Data Knowledge Graph built from fragmented telemetry with a Literature Knowledge Graph extracted from papers, reports, and traces, and then uses graph-guided prompting to constrain a LLM to domain vocabulary, units, and evidence-backed reasoning (Akewar et al., 10 Feb 2026).

Its ontology is formalized as

$u_{\rm g}$6

with classes such as EnvironmentalFactor, WorkloadAttribute, OperationType, and Metric, and the resulting graph as

$u_{\rm g}$7

Stage I constructs the Literature KG from unstructured SSD documents, storing typed claims in RDF/Turtle with provenance, confidence, and SPARQL access. Stage II transforms time-disjoint telemetry into an intermediate representation of typed frames — AttributeFrame, WorkloadFrame, EnvFrame, DataQuality, and Episode — and materializes these into a queryable Data KG. The framework defines “full-spectrum” analysis as descriptive, predictive, prescriptive, and what-if reasoning.

The evaluation reported in the paper is quantitative. On device-level SMART+Workload+Env analysis, KORAL achieved Precision $u_{\rm g}$8, Recall $u_{\rm g}$9, Accuracy $p_{\rm g}$0, TTF-MSE $p_{\rm g}$1 days, and Tail Latency MSE $p_{\rm g}$2, with prescriptive FiP up to $p_{\rm g}$3 and what-if CFV up to $p_{\rm g}$4. On fleet-level SMART+Workload analysis, it achieved Precision $p_{\rm g}$5, Recall $p_{\rm g}$6, Accuracy $p_{\rm g}$7, and TTF-MSE $p_{\rm g}$8 days, with prescriptive FiP $p_{\rm g}$9 and what-if CFV $$R^{\mu\nu}=\frac{4}{3}E_R u_R^\mu u_R^\nu+\frac{1}{3}E_R g^{\mu\nu},$$0. The paper presents this system as the first end-to-end combination of LLMs and KGs for full-spectrum SSD reasoning and releases the SSD-specific knowledge graph for reproducible research (Akewar et al., 10 Feb 2026).

In encyclopedic terms, the two KORALs are best understood as unrelated systems that share a name but belong to different technical lineages. The astrophysical KORAL is a GRRHD/GRRMHD code centered on conservative relativistic fluid and radiation evolution, whereas the SSD KORAL is a knowledge-orchestrated reasoning system centered on ontology alignment, graph materialization, and grounded LLM inference.