Physics Formulations

⚛️ Physics Formulations

GRANITE v0.6.8 | ← Parameter Reference | Roadmap →

Complete governing equations with references. For implementation details see docs/DEVELOPER_GUIDE.md §5.

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1. Spacetime: CCZ4 Formulation

GRANITE evolves the Einstein field equations G_μν = 8π T_μν using the CCZ4 (Conformal and Covariant Z4) formulation of Alic et al. (2012). CCZ4 extends BSSN by promoting the algebraic constraint Z_μ = 0 to a dynamical field, enabling exponential damping.

Conformal decomposition:

γ̃_ij = χ · γ_ij,   χ = (det γ_ij)^{-1/3}

Evolution equations:

∂_t χ = (2/3) χ (α K − ∂_i β^i) + β^i ∂_i χ

∂_t γ̃_ij = −2α Ã_ij + γ̃_ik ∂_j β^k + γ̃_jk ∂_i β^k
            − (2/3) γ̃_ij ∂_k β^k + β^k ∂_k γ̃_ij

∂_t K = −D_i D^i α + α(Ã_ij Ã^ij + K²/3)
         − κ₁(1+κ₂)αΘ + 4πα(ρ + S) + β^i ∂_i K

∂_t Θ = (1/2)α(R − Ã_ij Ã^ij + (2/3)K²)
          − α Γ̂^μ_{μν} Z^ν − κ₁(2+κ₂)αΘ + β^i ∂_i Θ

Production defaults: κ₁ = 0.02, κ₂ = 0, η = 2.0

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2. Gauge Conditions

1+log Slicing (Lapse)

∂_t α = β^i ∂_i α − 2α K

Singularity-avoiding: α → 0 near punctures, preventing evolution from reaching physical singularity.

Gamma-Driver (Shift) — Moving Punctures

∂_t β^i = (3/4) B^i
∂_t B^i = ∂_t Γ̃^i − η B^i

(Campanelli et al. 2006; Baker et al. 2006)

Trumpet geometry: At sufficient resolution (dx < 0.2M) and with ko_sigma = 0.1, the lapse profile forms a trumpet geometry near each puncture. At coarser resolution (dx > 0.5M), the trumpet is unresolved — this is expected behavior, not a bug.

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3. GRMHD: Valencia Formulation

The matter sector evolves conserved variables via:

∂_t U + ∂_i F^i(U) = S(U, g_μν)

Conserved variables:

U = √γ · (D, S_j, τ, B^i, D·Y_e)^T

D   = ρ W                               (baryon density)
S_j = ρ h W² v_j + ε_jkl (v^k − β^k/α) B^l √γ/α  (momentum)
τ   = ρ h W² − P − D                    (energy)

Geometric source terms S couple to spacetime via Christoffel symbols evaluated from the CCZ4 GRMetric3 output.

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4. Magnetic Field: Constrained Transport

Evolves via the induction equation:

∂_t B^i + ∂_j (v^j B^i − v^i B^j) = 0

subject to: ∂_i B^i = 0 (no magnetic monopoles)

GRANITE enforces ∇·B = 0 to machine precision at all times via Constrained Transport (Evans & Hawley 1988). The magnetic field is stored on cell faces (staggered grid); the CT update guarantees exact discrete divergence = 0.

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5. Equations of State

EOS Type	Implementation	Use Case
Ideal gas (Γ-law)	IdealGasEOS(Γ)	SMBH mergers (Γ=4/3), validation tests
Tabulated nuclear	TabulatedEOS with tri-linear interpolation in (ρ, T, Y_e)	Neutron star mergers (v0.8+)

Sound speed is always computed exactly from the EOS — never from a hardcoded Γ (fixed bug C1).

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6. Radiation: M1 Moment Closure (Built, Not Active in v0.6.8)

Energy equation:

∂_t E + ∂_i F^i = −κ_a (E − E_eq) + Q_leakage

Flux equation:

∂_t F^i + ∂_j P^ij = −(κ_a + κ_s) F^i

M1 Eddington tensor (Minerbo 1978, Levermore 1984):

P^ij = [(1−ξ)/2] δ^ij E + [(3ξ−1)/2] n^i n^j E

ξ = ξ(f),  f = |F|/(cE)  (radiation flux factor)

⚠️ Status: The M1 module is implemented and tested but is not called in the main RK3 evolution loop in v0.6.8. Radiation physics is completely inactive. Planned for v0.7.

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7. Numerical Methods

Spatial Derivatives (4th-order)

(∂_x f)_i = (−f_{i+2} + 8f_{i+1} − 8f_{i−1} + f_{i−2}) / (12 Δx) + O(Δx⁴)

Requires nghost = 4.

Kreiss-Oliger Dissipation

(∂_t u)_KO = −σ (Δx)^{2p-1} / 2^{2p} · D^{2p} u,   p = 3 (6th-order)

σ = 0.1 is the safe default. Never exceed 0.15.

SSP-RK3 (Shu & Osher 1988)

U^(1)   = U^n + Δt L(U^n)
U^(2)   = (3/4) U^n + (1/4) [U^(1) + Δt L(U^(1))]
U^(n+1) = (1/3) U^n + (2/3) [U^(2) + Δt L(U^(2))]

Reconstruction Schemes

Scheme	Order	Stencil	Use case
PLM (Piecewise Linear Method)	2nd	3 cells	Quick tests
PPM (Colella & Woodward 1984)	3rd	4 cells	Contact discontinuities
MP5 (Suresh & Huynh 1997)	5th	6 cells	Production default

Riemann Solvers

Solver	Physics	Notes
HLLE	GR-aware (uses actual GRMetric3)	Fixed C3; fast but diffusive
HLLD (Miyoshi & Kusano 2005)	All 7 MHD wave families	Preferred for MHD accuracy

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8. Full Reference List

Reference	Used in GRANITE
Alic et al. 2012, PRD 85, 064040	CCZ4 formulation
Baker et al. 2006, PRL 96, 111102	Moving punctures
Berger & Oliger 1984, J. Comput. Phys. 53, 484	AMR + subcycling
Bona et al. 1995, PRL 75, 600	1+log slicing
Brandt & Brügmann 1997, PRL 78, 3606	Two-Punctures ID
Campanelli et al. 2006, PRL 96, 111101	Moving punctures / Gamma-driver
Colella & Woodward 1984, J. Comput. Phys. 54, 174	PPM reconstruction
Evans & Hawley 1988, ApJ 332, 659	Constrained transport
Levermore 1984, J. Quant. Spectrosc. Radiat. Transfer 31, 149	M1 closure
Minerbo 1978, J. Quant. Spectrosc. Radiat. Transfer 20, 541	Eddington factor
Miyoshi & Kusano 2005, J. Comput. Phys. 208, 315	HLLD Riemann solver
Nakano et al. 2015, PRD 91, 104022	Ψ₄ extrapolation
Reisswig & Pollney 2011, CQG 28, 195015	Fixed-frequency GW integration
Shu & Osher 1988, J. Comput. Phys. 77, 439	SSP-RK3
Suresh & Huynh 1997, J. Comput. Phys. 136, 83	MP5 reconstruction

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See also: Architecture Overview | Benchmarks & Validation | Scientific Context

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