State of the Art MHD Methods for Astrophysical Applications p.1/32

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1 State of the Art MHD Methods for Astrophysical Applications Scott C. Noble February 25, 2004 CTA, Physics Dept., UIUC State of the Art MHD Methods for Astrophysical Applications p.1/32

2 Plan of Attack Is NOT a survey of all possible methods! Highlight relatively new methods! Focus on tests and comparisons between the methods. Cover general idea of algorithms. Details are deferred to given references. State of the Art MHD Methods for Astrophysical Applications p.2/32

3 Outline Introduction to ideal Relativistic MagnetoHydroDynamics (RMHD). Two basic approaches: Non-conservative and Conservative Methods for preserving the Divergence Constraint Conclusions State of the Art MHD Methods for Astrophysical Applications p.3/32

4 Ideal Relativistic Magnetohydrodynamics Perfect fluid = Isotropic, inviscid, no thermal conductivity: T µν fluid = (P + ρ + ρɛ) uµ u ν + P g µν µ (ρu µ ) = 1 g µ ( gρu µ ) = 0 Electrically neutral, infinitely conducting fluid: T µν EM = F µα F ν α 1 4 gµν F αβ F αβ µ F µν = J ν µ F µν = 0, F µν = 1 2 ε µνκλf κλ Infinite conductance F µν u µ = 0 State of the Art MHD Methods for Astrophysical Applications p.4/32

5 Ideal RMHD (cont.) ν ν = = i t F µν u µ = 0 µ F µν = 0 µ F µν = J ν v j = u i /u t b µ F µν u ν = 1 2 εµνκλ F κλ u ν F µν = b µ u ν b ν u µ b µ u µ = 0 B i F it b t = B i u µ g iµ b i = 1 u t `Bi + b t u i t ` gb i = j ˆ g `Bj v i B i v j j ` gb j = 0 State of the Art MHD Methods for Astrophysical Applications p.5/32

6 Ideal RMHD (cont.) ν ν = = i t F µν u µ = 0 µ F µν = 0 µ F µν = J ν v j = u i /u t b µ F µν u ν = 1 2 εµνκλ F κλ u ν F µν = b µ u ν b ν u µ b µ u µ = 0 B i F it b t = B i u µ g iµ b i = 1 u t `Bi + b t u i t ` gb i = j ˆ g `Bj v i B i v j j ` gb j = 0 State of the Art MHD Methods for Astrophysical Applications p.5/32

7 Ideal RMHD (cont.) ν ν = i t F µν u µ = 0 µ F µν = 0 µ F µν = J ν v j = u i /u t b µ F µν u ν = 1 2 εµνκλ F κλ u ν F µν = b µ u ν b ν u µ b µ u µ = 0 B i F it b t = B i u µ g iµ b i = 1 u t `Bi + b t u i t ` gb i = j ˆ g `Bj v i B i v j j ` gb j = 0 State of the Art MHD Methods for Astrophysical Applications p.5/32

8 Ideal RMHD (cont.) ν ν = i t F µν u µ = 0 µ F µν = 0 µ F µν = J ν v j = u i /u t b µ F µν u ν = 1 2 εµνκλ F κλ u ν F µν = b µ u ν b ν u µ b µ u µ = 0 B i F it b t = B i u µ g iµ b i = 1 u t `Bi + b t u i t ` gb i = j ˆ g `Bj v i B i v j j ` gb j = 0 State of the Art MHD Methods for Astrophysical Applications p.5/32

9 Ideal RMHD (cont. II) T µν MHD = `ρ + ρɛ + P + b 2 u µ u ν + `P b2 δ µ ν b µ b ν µ T µ ν = 0 t g ρu t T t µ B i j g 6 4 ρu j T j µ B j v i B i v j = g T α βγ β αµ t ( gq) + j ` gf j = S(q, g µν ) j ` gb j = 0 State of the Art MHD Methods for Astrophysical Applications p.6/32

10 Non-conservative Methods t q + i f i = S(q, µ q) Experienced methods from hyrdro. simulations; Excellent for smooth flows; Computationally efficient; Use of upwind, monotonic methods for advection helps w/ shocks; Requires use of artificial viscosity, P P + Q; Still problems near shocks, especially as v 1; Norman and Winkler, in Astrophys. Radiation Hydro., 449 (1986) State of the Art MHD Methods for Astrophysical Applications p.7/32

11 ZEUS-like: FD/CT/MOC-CT Methods FD = Finite Difference, MOC = Method of Characteristics CT = Constrained Transport Wilson, in 7th Texas Symp. on Rel. Astrophys., [MHD, Schwarz. Accretion] Hawley, Smarr, Wilson, ApJ 277, 296 (1984). [Hydro. Eq.s] Hawley, Smarr, Wilson, ApJS 55, 211, (1984). [Hydro. Tests, Kerr Accretion] Evans and Hawley, ApJ 332, 659 (1988). [CT, 2D MHD, Schwarzschild Accretion] Stone and Norman, ApJS 80, 791 (1992). [ZEUS-2D, MOC-CT] ZEUS Code webpage, jstone/zeus.html Hawley, Stone, Comput. Phys. Commun. 89, 127 (1995). [MOC-CT] De Villiers and Hawley, ApJ 589, 458 (2003). [3D MHD, Kerr Accretion] State of the Art MHD Methods for Astrophysical Applications p.8/32

12 De Villiers and Hawley 2003 (DH) Non-conservative scheme: i t q + i hv i G(q, g µν ) = S(q, µ q, g µν, α g µν ) Evolution is broken down into several sub-steps: t q = L(q) MOC CT : Finds b µ q 1 = q 0 + tl 0 (q 0 ) : Transport L 0 i ( γv i H), H = {D, E, S j } q 2 = q 1 + tl 1 (q 1 ) : Source L 1 µ {W, P, g νλ, b ν }, q Staggered mesh: vectors at cell faces, scalars at cell centers; New version of MOC-CT for i B i = 0; Full 3D MHD in Boyer-Lindquist coordinates; L 0 found via monotonic upwind scheme (van Leer 1977); State of the Art MHD Methods for Astrophysical Applications p.9/32

13 Conservative Methods: Finite Volume t q + x f = S(q) q n+1 j q n j = t `F j+1/2 F j 1/2 n+1/2 + t S n+1/2 j n+1 q j n+1 F n+1/2 j 1/2 F n+1/2 j+1/2 n q j n State of the Art MHD Methods for Astrophysical Applications p.10/32

14 Conservative Methods: Godunov Schemes t q + i f i = S(q) Assume piecewise-constant data, q q State of the Art MHD Methods for Astrophysical Applications p.11/32

15 Conservative Methods: Godunov Schemes t q + i f i = S(q) Assume piecewise-constant data, q q For higher resolution, interpolate for q L and q R State of the Art MHD Methods for Astrophysical Applications p.11/32

16 Conservative Methods: Godunov Schemes t q + i f i = S(q) Assume piecewise-constant data, q q For higher resolution, interpolate for q L and q R Need to use monotonic (aka slope-limiting) interpolation; State of the Art MHD Methods for Astrophysical Applications p.11/32

17 Conservative Methods: Godunov Schemes t q + i f i = S(q) Assume piecewise-constant data, q q For higher resolution, interpolate for q L and q R Need to use monotonic (aka slope-limiting) interpolation; q L q R State of the Art MHD Methods for Astrophysical Applications p.12/32

18 Riemann Problem q L q R Find q(x, t) given t q + x f = 0 8 < q L for x < 0 q(x, t = 0) = : q R for x > 0 Riemann solution F n+1/2. State of the Art MHD Methods for Astrophysical Applications p.13/32

19 Hydrodynamic Riemann Solution Rarefaction Fan L * P *, v *, ρo Contact Discontinuity * R ρ * P, v *, o Shock v L L P L,, ρ o R R R P, v, ρo State of the Art MHD Methods for Astrophysical Applications p.14/32

20 MHD Riemann Solution F L A L S L C S R A R F R L q R q F L,R = Fast magnetosonic wave S L,R = Slow magnetosonic wave A L,R = Alfvenic wave C = Contact Discontinuity State of the Art MHD Methods for Astrophysical Applications p.15/32

21 Approximate Riemann Solvers: HLL t q + A i i q = 0 H L q* H R L q R q F n+1/2 j+1/2 = 8 >< >: 1 (H R H L ) f(q L ) ˆHR f L H L f R + H L H R `q R q L if 0 H L if H L 0 H R f(q R ) if H R 0 State of the Art MHD Methods for Astrophysical Applications p.16/32

22 Approximate Riemann Solvers: HLLC H* H L q L* R* q H R L q R q F n+1/2 j+1/2 = 8 >< >: f(q L ) if 0 H L f(q L ) + H `q L L q L if H L 0 H f(q R ) + H `q R R q R if H L 0 H f(q R ) if H R 0 State of the Art MHD Methods for Astrophysical Applications p.17/32

23 Magnetized Bondi Flow HARM DH 2003 State of the Art MHD Methods for Astrophysical Applications p.18/32

24 Gammie Inflow Steady-state, equatorial, magnetized inflow in Kerr inside marginally stable orbit; HARM B φ DH 2003 State of the Art MHD Methods for Astrophysical Applications p.19/32

25 Slow Shock Komissarov, MNRAS 303, 343 (1999). DH 2003 State of the Art MHD Methods for Astrophysical Applications p.20/32

26 Rarefaction Komissarov N = 150 DH 2003 N = 2048 State of the Art MHD Methods for Astrophysical Applications p.21/32

27 Qualities of Conservative GRMHD Schemes Improved behavior in shock tests Equivalent behavior in relativistic cases Better behavior for ultra-relativistic flows Still developmental Worse behavior in floored regions Worse behavior when P mag P Conservative Primitive variable calculation Slower code! State of the Art MHD Methods for Astrophysical Applications p.22/32

28 Other Schemes Del Zanna et al., AA 400, 397 (2003). 3rd-order CENO with HLL SRMHD Balsara, ApJS 132, 83 (2001). Roe-type Approx. Riemann Solver Interpolates characteristic variables SRMHD ATHENA, jstone/athena.html Roe-type Approx. Riemann Solver Not relativistic State of the Art MHD Methods for Astrophysical Applications p.23/32

29 Maintianing the Divergence Constraint Toth, J. Comp. Phys., 161, 605 (2000). TVD-Lax-Friedrich i B i 0 leads to: Non-perpendicular Lorentz forces to B i Inconsistency in MHD Instabilities and spurious effects E z = v x B y v y B x = f x CT : Evans and Hawley 1988 MOC-CT: Hawley and Stone 1995 MOC-CT2: De Villiers and Hawley 2003 State of the Art MHD Methods for Astrophysical Applications p.24/32

30 Flux/Field CT/CD for Conserv. Methods Field-CT: Interp. predicted B i and v i for E n+1/2 j+1 ; State of the Art MHD Methods for Astrophysical Applications p.25/32

31 Flux/Field CT/CD for Conserv. Methods Field-CT: Interp. predicted B i and v i for E n+1/2 j+1 ; Flux-CT: Interp. f i and f in for E n+1/2 j+1/2 ; State of the Art MHD Methods for Astrophysical Applications p.25/32

32 Flux/Field CT/CD for Conserv. Methods Field-CT: Interp. predicted B i and v i for E n+1/2 j+1 ; Flux-CT: Interp. f i and f in for E n+1/2 Field-CD: E n+1/2 j = 1 2 `E n j + Ej ; j+1/2 ; State of the Art MHD Methods for Astrophysical Applications p.25/32

33 Flux/Field CT/CD for Conserv. Methods Field-CT: Interp. predicted B i and v i for E n+1/2 j+1 ; Flux-CT: Interp. f i and f in for E n+1/2 Field-CD: E n+1/2 j = 1 2 `E n j Flux-CD: f i for E n+1/2 j ; + Ej ; j+1/2 ; State of the Art MHD Methods for Astrophysical Applications p.25/32

34 8-Wave Formulation K. G. Powell, P. L. Roe, et al., J. Comput. Phys. 154, 284 (1999). K. G. Powell, ICASE Report No , Langley, VA, MHD equations can be derived w/o assuming constraint source terms i B i ; With i B i source terms, conserves divergence constraint along flow lines: t ` i B i + `vj j i B i = 0 Supposed to preserve initially constrained data along flows; Any monopoles are transported away hopefully off the grid; Increases robustness, G. Tóth and D. Odstrěil, J. Comput. Phys. 128, 82 (1996). Formulation becomes non-conservative problems near strong shocks; State of the Art MHD Methods for Astrophysical Applications p.26/32

35 Divergence Cleaning Schemes J. U. Brackbill and D. C. Barnes, J. Comput. Phys. 35, 426 (1980). C.-D. Munz, J. Comput. Phys. 161, 484 (2000). A. Dedner, et al., J. Comput. Phys. 175, 645 (2002). S. S. Komissarov, Appendix C of astro-ph/ t B i j v i B j v j B i + g ij Ψ = 0 D(Ψ) + i B i = 0 t D(Ψ) = i i Ψ D(Ψ) = 0 i i Ψ = 0 Elliptic D(Ψ) = 1 p Ψ tψ = p i i Ψ Parabolic D(Ψ) = 1 h 2 t Ψ tt Ψ = h 2 i i Ψ Hyperbolic D(Ψ) = 1 h 2 t Ψ + 1 p 2 Ψ tt Ψ + h2 p 2 t Ψ = h 2 i i Ψ Mixed State of the Art MHD Methods for Astrophysical Applications p.27/32

36 Discretization Differences in Constraint State of the Art MHD Methods for Astrophysical Applications p.28/32

37 2D Shock Tube Test State of the Art MHD Methods for Astrophysical Applications p.29/32

38 2D Shock Tube Test State of the Art MHD Methods for Astrophysical Applications p.30/32

39 Orzag-Tang Vortex State of the Art MHD Methods for Astrophysical Applications p.31/32

40 State of the Art MHD Methods for Astrophysical Applications p.32/32