Komatsu-Eriguchi-Hachisu Differential Rotation Law

• Komatsu-Eriguchi-Hachisu Differential Rotation Law is a formula characterizing differential rotation in relativistic systems, vital for describing rapidly rotating neutron stars.
• It integrates a Newton–Raphson spectral solver with modular domain decomposition and spectral basis functions to achieve exponential convergence in complex geometries.
• The framework’s robust design supports diverse astrophysical scenarios and facilitates direct interfacing with dynamical evolution codes for comprehensive compact object modeling.

The Frankfurt University/KADATH (FUKA) Suite is a publicly available computational framework for generating high-accuracy initial data for a range of relativistic astrophysical systems, with a particular focus on stationary, axisymmetric equilibrium models of differentially rotating neutron stars within full general relativity. The suite leverages the KADATH spectral library for multidomain pseudospectral solutions and includes modular architectures for domain decomposition, equation-of-state integration, and multidimensional elliptic solves. Its extensibility encompasses isolated stellar configurations, binaries (BNS, BBH, BHNS), alternative rotation laws, magnetic fields, and microphysical treatments. Solver outputs within FUKA are designed for direct interfacing with contemporary dynamical evolution codes in the field.

1. Suite Architecture and Component Integration

The FUKA suite is organized around a high-level modular architecture built atop KADATH’s robust spectral and domain-management capabilities (Tootle et al., 8 Jan 2026, 0909.1228). The key architectural layers include:

• Geometry Layer: Abstracts coordinate systems and domains, supporting spherical, bispherical, and cylindrical-critical geometries. Domain classes (Domain_nucleus, Domain_shell, etc.) provide mappings between physical and numerical coordinates and are responsible for enforcing matching at domain boundaries and handling singularities.
• Spectral Basis and Collocation: Enables choice of Chebyshev or Legendre decompositions for radial direction, trigonometric or Legendre for angular directions, and Fourier modes for azimuthal dependence. Spectral regularity is ensured via Galerkin bases and domain adaptation, with collocation points placed strategically to maximize convergence and mitigate spectral Gibbs phenomena at surfaces.
• System Assembly & Solver Layer: Facilitates equation declaration using LaTeX-inspired syntax, variable and field registration, and aggregation into a global residual vector. The solver employs a Newton–Raphson approach with automatic differentiation and residual-based stopping criteria (ε_STOP ≈ 10⁻⁸), supported by LAPACK or ScaLAPACK for parallel linear algebra.

Integration with KADATH enables FUKA to transparently import spectral elements, boundary conditions, and parallelization strategies, with I/O routines adapted for spectral data export in HDF5 format. Installation leverages CMake-based build processes and links KADATH as an external solver module.

2. Governing Equations and Spectral Decomposition

FUKA targets the general-relativistic constraint equations for stationary, axisymmetric systems, with two principal solver modules:

• Quasi-Isotropic Coordinate (QIC) Solver: Uses (t, r, θ, φ) coordinates and metric

$$g_{\mu\nu}dx^\mu dx^\nu = -\alpha^2 dt^2 + A^2 (dr^2 + r^2 d\theta^2) + B^2 r^2 \sin^2\theta (d\phi + \beta^\phi dt)^2$$

with scalar unknowns ν(r,θ), ω(r,θ), A(r,θ), B(r,θ). The reduced Einstein equations take the form of coupled Laplacians (Δ₃, Δ₂ operators), populated by fluid and metric source terms derived from the energy-momentum tensor of an isentropic, axisymmetric fluid.

• Extended Conformal Thin Sandwich (XCTS) Solver: Operates on 3D Cartesian collocation grids, seeking spectral solutions for the conformal factor Ψ, shift vector β^i, and product αΨ (lapse × conformal). The governing elliptic system enforces conformal flatness, maximal slicing (K=0), and stationarity through coupled equations for Ψ, β^i, and αΨ, with fluid sources handled analogously to the QIC formulation.

Both solvers adopt the Komatsu–Eriguchi–Hachisu (KEH) differential rotation law:

$$\Omega(r, \theta) = \frac{\Omega_c}{1 + \hat A^2 r^2 \sin^2\theta}$$

and integrate the Euler equation via a hydrostationary first integral:

$$\ln \alpha + \ln h - \ln W - \frac{1}{2}\frac{j^2}{A^2} = C$$

Spectral domains are constructed as a central nucleus, adapted surface-matched shells, and an outer compactified shell mapping $r \to \infty$ to a finite coordinate.

3. Equation-of-State and Microphysical Support

FUKA provides robust EOS integration, supporting:

• Piecewise-polytropic EOS forms
• Cold tabulated EOS in LORENE format
• 3D tabulated "stellar-collapse" EOS tables via GRHayLEOS (functions of ρ, T, Y_e). While current implementations restrict models to cold beta-equilibrium slices (h = h(ρ), ds = 0), the infrastructure is extensible to finite-temperature, non-isentropic treatments.

Boundary conditions at $r \to \infty$ enforce weak-field flatness, while adapted shell interfaces demand regularity and matching of fields and derivatives.

4. Numerical Implementation and Validation

The Newton–Raphson spectral approach in FUKA displays exponential convergence up to effective spectral resolutions of $\tilde N \sim 20-25$, beyond which steep gradients and floating-point precision set error floors. Validation is performed through:

• Self-Convergence: Diagnostics such as ADM mass, Komar mass, baryonic mass, and central density show exponential error decay, with minima at $\sim10^{-8}$ in residuals for $\tilde N \simeq 22$.
• QIC vs. XCTS Comparisons: Axisymmetric configurations (e.g., SB6, U13 models) yield ADM and Komar masses agreeing to $\sim10^{-8}$, and baryonic masses within $\sim10^{-3}$.
• Code Cross-Validation: For sequences of $\hat A = \{0.5, 1, 2\}$ and axis ratios $r_p/r_e = \{0.4, 0.6, 0.8, 1.0\}$, QIC-derived $M_b(\rho_c)$ curves highly resemble those of the public RNS code, with minor discrepancies at extreme rotation.
• Spectral Regularity: No spectral Gibbs phenomenon arises at the stellar surface due to interface-matching of domains and bases regular at singularities.

Table 1. Select Validation Results

Diagnostic	QIC vs. XCTS Agreement	QIC vs. RNS Agreement
ADM mass	~10⁻⁸	few percent
Komar mass	~10⁻⁸	few percent
Baryonic mass, central dens.	~10⁻³	few percent

5. Performance, Dynamical Evolution, and Error Analysis

Performance metrics indicate that QIC runs at moderate spectral resolutions (R ≈ 17) complete in minutes on quad-core laptop environments with ~16 GB RAM. Unassisted XCTS solves can extend to hours or days, but the hybrid workflow (QIC initialization interpolated to XCTS collocation) reduces wall time for the latter to minutes.

In dynamical context (Einstein Toolkit + IllinoisGRMHD + BSSN), evolutions of bar-stable/unstable models (SB6, U13) demonstrate second-order convergence in Hamiltonian constraint violations, congruent with the finite-difference evolution scheme. Error analyses reveal that, for initial data at R ≳ 13, the dynamical scheme resolution (Δx = 0.32 vs 0.40) overtakes initial data resolution as the dominant error source. Bar-mode instability diagnostics (m=2 Fourier amplitude) remain consistent to within ~10% across resolutions.

6. Extensibility and Future Prospects

The KADATH infrastructure within FUKA renders the suite highly extensible. Supported and planned enhancements include:

• Alternative rotation laws (Uryu et al., multi-parameter families, Cam­elio models)
• Non-isentropic fluids (relaxing ds=0 constraint, solving for temperature T)
• Magnetic fields via Maxwell stress tensor inclusion
• Full 3D, non-axisymmetric equilibrium solutions and binary initial data generation
• Integration of neutrino transport and microphysics via GRHayLEOS/GRHayL
• Direct export compatibility with evolution codes (Einstein Toolkit, SpECTRE, BAM, SACRA, SPHINCS_BSSN)

All codes, documentation, and examples are hosted on public repositories, with workflow adaptation to contemporary research codes in gravitational-wave astrophysics and stellar collapse. The suite’s detailed implementations and validations position it as a cornerstone resource for relativistic stellar and compact-object initial data studies (Tootle et al., 8 Jan 2026, 0909.1228).