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Radiation Magnetohydrodynamics (Rad-MHD) 코드 개발 I

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By Donghui Son 6 min read

[Study Note] Rad-MHD: FLD approximation vs M1 closure

설명용 blog 문체는 이번 글에서 버립니다.
이 문서는 내가 구현하면서 다시 보는 working note입니다.

핵심 키워드 네 개만 고정:

  • sign convention
  • asymptotic limit
  • stiffness
  • realizability

0) Scope (이번 note에서 다루는 것)

  • non-relativistic single-fluid MHD
  • grey radiation approximation (frequency-integrated)
  • FLD approximation
  • M1 moment closure
  • numerical implementation 관점의 체크포인트

1) 먼저 고정할 convention

magnetic-field unit convention은 $4\pi=1$.
variables:

  • $\rho$: mass density
  • $\mathbf{v}$: fluid velocity
  • $P$: gas pressure
  • $\mathbf{B}$: magnetic field
  • $E$: total energy density (internal + kinetic + magnetic)
  • $E_r, \mathbf{F}_r, \mathbb{P}_r$: radiation energy density, radiation flux, radiation pressure tensor
  • $G^\mu=(G^0,\mathbf{G})$: radiation four-force density on matter

sign convention:

  • matter equations: $+\mathbf{G}, +cG^0$
  • radiation equations: $-\mathbf{G}, -cG^0$

이 부호를 처음에 고정하지 않으면 뒤에서 계속 꼬입니다.

2) Governing equations (baseline card)

  • Mass conservation
\[\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{v}) = 0\]
  • Momentum conservation
\[\frac{\partial (\rho \mathbf{v})}{\partial t} + \nabla \cdot \left[ \rho \mathbf{v}\otimes\mathbf{v} + \left(P + \frac{B^2}{2}\right)\mathbb{I} - \mathbf{B}\otimes\mathbf{B} \right] = \mathbf{G}\]
  • Total energy conservation (ideal-MHD form)
\[\frac{\partial E}{\partial t} + \nabla \cdot \left[ \left(E + P + \frac{B^2}{2}\right)\mathbf{v} - \mathbf{B}(\mathbf{v}\cdot\mathbf{B}) \right] = cG^0\] \[E = \rho e + \frac{1}{2}\rho v^2 + \frac{B^2}{2}\]
  • Induction equation (ideal MHD)
\[\frac{\partial \mathbf{B}}{\partial t} = \nabla \times (\mathbf{v}\times\mathbf{B})\]
  • Resistive extension
\[\frac{\partial \mathbf{B}}{\partial t} = \nabla \times \left(\mathbf{v}\times\mathbf{B} - \eta \nabla\times\mathbf{B}\right)\]

resistive MHD를 쓰면 energy equation에 Ohmic heating term도 같이 관리해야 합니다.

  • Divergence constraint
\[\nabla\cdot\mathbf{B}=0\]
  • Radiation moment equations
\[\frac{\partial E_r}{\partial t} + \nabla\cdot\mathbf{F}_r = -cG^0\] \[\frac{1}{c^2}\frac{\partial \mathbf{F}_r}{\partial t} + \nabla\cdot\mathbb{P}_r = -\mathbf{G}\]
  • Equation of state (ideal gas)
\[P=(\gamma-1)\rho e,\qquad T=\frac{\mu m_H}{k_B}\frac{P}{\rho}\]

3) RSLA memo

Reduced Speed of Light Approximation (RSLA)는 radiation transport time scale을 줄여서 cost를 낮춥니다.

  • wave speed를 정하는 $c$: $\hat{c}\ll c$로 대체 가능
  • matter-radiation coupling source term의 물리적 $c$: 보통 유지

coupling term까지 같이 줄이면 equilibrium temperature와 coupling time scale이 왜곡될 수 있습니다.
RSLA는 trick이 아니라 error budget 선택입니다.

4) FLD approximation note

FLD는 radiation field를 diffusion limit로 처리합니다.

\[\mathbf{F}_r = -\frac{c\lambda(R)}{\kappa_R \rho}\nabla E_r,\qquad R=\frac{|\nabla E_r|}{\kappa_R\rho E_r}\]

필수 limit:

  • $R\ll1$: $\lambda\to 1/3$ (diffusion limit)
  • $R\gg1$: $\lambda\to 1/R$ (free-streaming limit) $\mathbf{F}_r\le cE_r$

대표 flux limiter:

  • Levermore-Pomraning: \(\lambda(R)=\frac{2+R}{6+3R+R^2}\)
  • Minerbo: \(\lambda(R)=\frac{1}{R}\left(\coth R-\frac{1}{R}\right)\)

grey-LTE form:

\[\frac{\partial E_r}{\partial t} - \nabla\cdot\left(\frac{c\lambda(R)}{\kappa_R\rho}\nabla E_r\right) = c\kappa_P\rho(a_rT^4-E_r)\]

Non-LTE에서는 $\kappa_P,\kappa_E,\kappa_A$를 동일값으로 두지 않습니다.

FLD 한 줄 요약:

  • Pros: 단순하고 optically thick regime에서 안정적
  • Cons: angular fidelity가 낮아 shadowing, multi-beam crossing에서 약함

5) M1 closure note

M1 closure는 Radiation transfer equation의 angular moment system을 닫는 방법입니다.

\[\mathbb{P}_r=\mathbb{D}E_r\] \[\mathbb{D} =\frac{1-\chi}{2}\mathbb{I} +\frac{3\chi-1}{2}\mathbf{n}\otimes\mathbf{n},\qquad \mathbf{n}=\frac{\mathbf{F}_r}{|\mathbf{F}_r|}\] \[f=\frac{|\mathbf{F}_r|}{cE_r},\qquad \chi(f)=\frac{3+4f^2}{5+2\sqrt{4-3f^2}}\]

필수 제약:

  • $0\le f\le1$
  • $f\to0 \Rightarrow \chi\to1/3$
  • $f\to1 \Rightarrow \chi\to1$

$f=0$에서는 $\mathbf{n}$이 정의되지 않으므로 implementation에서 $\mathbb{D}=\mathbb{I}/3$ limit를 직접 사용합니다.

M1 한 줄 요약:

  • Pros: FLD보다 angular transport를 더 잘 반영
  • Cons: computational cost 증가, multi-beam crossing에서 구조적 오차 잔존

정확도는 공짜가 아닙니다. M1은 그 대가를 equation count로 냅니다.

6) FLD vs M1 quick selection table

MetricFLDM1
Dominant optical-depth regimeoptically thick dominatedthin/thick coexistence
Angular fidelitylowmedium to high
Computational costlowhigh
Shadow/beam handlinglimitedpartially available
Multi-beam crossingweakstill weak
Initial implementation complexitylowhigh

내 기준 workflow:

  1. FLD로 먼저 scale/sensitivity 확인
  2. angular effect가 결과를 바꾸면 M1으로 전환
  3. 필요하면 region-wise hybrid strategy 검토

7) Numerical implementation checklist

7.1 Common components

  • Spatial discretization: conservative finite-volume method (FVM)
  • MHD flux: HLL/HLLD + high-order reconstruction (MUSCL, PPM, WENO)
  • $\nabla\cdot\mathbf{B}=0$: CT (Constrained Transport) or divergence cleaning
  • Time integration: explicit + implicit (IMEX)
  • Splitting: Strang splitting은 편하지만 splitting error는 항상 감안

7.2 FLD implementation core

FLD는 nonlinear diffusion이라 결국 large sparse linear algebra 문제로 바뀝니다.

\[\frac{E_r^{n+1}-E_r^n}{\Delta t} - \nabla\cdot\left(D(E_r^{n+1})\nabla E_r^{n+1}\right) = S(E_r^{n+1},T^{n+1})\]
  • Nonlinear solve: Picard or Newton-Krylov
  • Linear solver: GMRES/BiCGSTAB + AMG/ILU preconditioner
  • Stiff source terms: fully implicit or semi-implicit

7.3 M1 implementation core

M1은 hyperbolic system이라 Godunov-type method가 기본 선택입니다.

  1. reconstruction으로 face left/right state 계산
  2. approximate Riemann solver로 radiation flux 계산
  3. conservative update
  4. stiff coupling source term 별도 처리 (implicit or semi-implicit)
  5. realizability correction: $E_r\ge0$, $\mathbf{F}_r\le cE_r$ 강제

realizability correction을 빼면 계산은 빨라져도 해는 물리가 아닙니다.

7.4 Minimum verification set

코드가 돌아간다고 물리가 맞는 건 아닙니다.

  • Marshak wave (diffusion verification)
  • radiative shock (stiff coupling verification)
  • shadow test (angular fidelity verification, FLD/M1 차이 확인)
  • beam propagation + beam crossing (M1 limitation 확인)

8) 결론 (내 메모용)

FLD와 M1은 둘 다 compromise입니다. 차이는 compromise 위치입니다.

  • FLD: cost 절약, angular information 손실
  • M1: angular information 확보, cost/complexity 증가

정답은 방법 이름이 아니라 problem setup에서 결정됩니다.
optical-depth structure, target observable, wall-clock budget을 먼저 정하고 선택하는 편이 빠릅니다.

참고문헌

  • Levermore, C. D. (1984). Relating Eddington factors to flux limiters. Journal of Quantitative Spectroscopy and Radiative Transfer, 31(2), 149-160.
  • Levermore, C. D., & Pomraning, G. C. (1981). A flux-limited diffusion theory. The Astrophysical Journal, 248, 321-334.
  • Mihalas, D., & Mihalas, B. W. (1984). Foundations of Radiation Hydrodynamics. Oxford University Press.
  • Turner, N. J., & Stone, J. M. (2001). A module for radiation hydrodynamic calculations with ZEUS-2D using flux-limited diffusion. The Astrophysical Journal Supplement Series, 135(1), 95.
  • Gonzalez, M., Audit, E., & Huynh, P. (2007). HERACLES: a three-dimensional radiation hydrodynamics code. Astronomy & Astrophysics, 464(2), 429-435.
Computational MHD, Radiation MHD
radiative-mhd numerical-methods fld-approximation m1-closure code-development advanced
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