Multi-D MHD and B = 0

Transcription

1 CapSel DivB - 01 Multi-D MHD and B = 0 keppens@rijnh.nl multi-d MHD and MHD wave anisotropies dimensionality > 1 non-trivial B = 0 constraint even if satisfied exactly t = 0: can numerically generate B 0 due to non-linearities of shock-capturing numerical methods B 0 build-up numerical instability (+ physical nonsense) need to control this somehow strategies for B = 0

2 CapSel DivB - 02 MHD wave signals locally (δ-function) perturb homogeneous magnetized plasma at rest take γ = 5/3, ρ = 1, p th = 0.6 and B = 0.9ê x (c = 1, b = 0.9) simulate on (x, y) [ 0.5, 0.5] 2 in 2.5D MHD (include v z, B z ) perturb at origin with δρ = 0.1, δv z = 0.01 and δp th = 0.06 MHD counterpart of throwing a stone in a puddle entropy, total pressure, B z at finite time

3 CapSel DivB waves: return in Friedrichs group diagram (theoretical one overplotted) group velocity for wave package (and energy flow direction) v gr = ω(k) k = ω ê x + ω ê y + ω ê z k x k y k z using wave frequency ω and wave vector k at origin: local entropy (density) disturbance remains (plasma at rest) slow magnetosonic wave signals: cusp-like features: anisotropic! fast magnetosonic waves: like sound waves (would be spherical signal) velocity c = γp/ρ along B [generally max(c, b = B/ ρ)] travel faster in perpendicular direction b 2 + c 2 Alfvén waves: extremely anisotropic: pure point-like along B travel at Alfvén speed b and sample magnetic field lines! note different polarizations: s only shows entropy wave, p shows slow and fast, Alfvén only visible in v z, B z

4 CapSel DivB - 04 B = 0 physically: only exact solenoidal field is allowed numerics: discretization error + machine precision unavoidable conservative form of momentum equation: uses divergence of Maxwell stress tensor equal to Lorentz force IF B = 0 since ) (I B2 2 B B = ( B ) B B ( B ) force orthogonal to B if solenoidal field would like discrete B = 0, orthogonal Lorentz force, conservative form difficult to satisfy all demands simultaneously but possible, cfr. Tóth JCP 182, 346 (2002)!!!

5 CapSel DivB - 05 Vector potential rewrite MHD equations in terms of vector potential A defined from B = A keeps B = 0 exactly analytically still need discrete (.) = 0 increases order of occuring spatial derivatives (accuracy loss) Boundary Conditions on A not always straightforward conflicts with current Roe-type solvers exploiting B

6 CapSel DivB - 06 Projection scheme should be sufficient to control numerical value of B enforce constraint in particular discretization to given accuracy Take corrective action by projection scheme [Brackbill & Barnes 1980] scheme yields B with B 0 correct to solenoidal B = B φ, solve 2 φ = B Poisson eqn., solve by (iterative) scheme up to desired accuracy projects B on subspace of zero divergence solutions no change in current density accuracy: does not need to be machine precision!

7 CapSel DivB - 07 does not violate conservation properties indirectly affects local thermal/magnetic balance in total energy same ideas in use for incompressible HD (where v = 0) uniform Cartesian grids: projection scheme makes smallest possible correction to remove divergence of B provided by base scheme given B, find closest field B from minimizing B B 2 under solenoidal constraint B = 0 does not affect order of accuracy of base scheme (remains same) proved by Tóth (2000): scheme is consistent for weak solutions Can use projection to eliminate finite divergence of discrete initial B

8 CapSel DivB - 08 Powell source terms solution is ok up to truncation error: same goes for B = 0 maintaining constraint to truncation error is sufficient system of 8 PDEs for ideal MHD equations in conservation form not Galilean invariant 8 8 system has eigenvalues zero, and v, v ± c s, v ± b, v ± c f spurious eigenvalue which conflicts with constraint B = 0 carries a jump in normal (to cell edge) component of B restore Galilean invariance by writing system as U t + ( F) = S sources S = (S ρ, S ρv, S e, S B ) proportional to B

9 CapSel DivB - 09 Powell suggests following sources: 0 ( B )B ( B )B v ( B )v restore Galilean invariance, replace zero with extra v eigenvalue introduces 8-wave approximate Riemann solver induction equation with source term B + (vb B v) = v( B ) t equivalent to evolution equation for B given by B + (v( B )) = 0 t passively convected scalar field B /ρ

10 CapSel DivB - 10 idea is that numerical divergence errors get advected away potential problems at internal flow stagnation points sources destroy conservation form potential violation of Rankine Hugoniot across shock case with incorrect jumps across discontinuities in Tóth (2000) usually does work correctly though! can be used with non-riemann solver based method (e.g TVDLF) restore conservation of momentum/energy by only taking S B along arguments given by Janhunen (2000) and Dellar (2001) Lorentz (instead of Galilean) invariance makes momentum and energy equations conservative

11 CapSel DivB - 11 Constrained Transport enforce B = 0 in one particular discretization kept zero to machine precision in one discretization must have initial field with zero B in chosen discretization must have BCs compatible with zero B in chosen discretization typical CT approaches employ staggered magnetic field representation with ρ, e at cell center, take B at cell edges fluxes in induction eqn at cell vertex (corners), B cell centered Tóth (2000): CT recast in FV sense: no staggering needed! VAC: 5 different variants of CT/central difference type schemes Tóth (2000): two new finite volume CD type schemes!

12 CapSel DivB - 12 Evans & Hawley CT idea place magnetic field components on cell interfaces 2D: take electric field E z Ω = (v B ) z at cell corners Ω b y Ω b x.b b x Ω b y Ω update cell interface B = (B x, B y ) from induction equation as B x,n+1 j+1/2,k Bx,n j+1/2,k t B y,n+1 j,k+1/2 By,n j,k+1/2 t = Ω j+1/2,k+1/2 Ω j+1/2,k 1/2 y = + Ω j+1/2,k+1/2 Ω j 1/2,k+1/2 x

13 CapSel DivB - 13 then B n+1 = 0 if B n = 0 with ( B ) j,k = Bx j+1/2,k Bx j 1/2,k x + By j,k+1/2 By j,k 1/2 y up to accuracy of round-off errors (machine precision) note: CT does not ensure B = 0 in any other discretization!

14 CapSel DivB - 14 variations of CT idea without staggering use spatio-temporal interpolations from cell-centered v and B B y v xb B y v xb Ω b y Ω v B b x v B b x B x B x B B x Ω b y Ω v B v B v xb B v xb y Field interpolated CT (spatial + temporal interpolation) versus Field interpolated Central Difference (latter uses only temporal interpolation) can both be written as FV conservative update on cell-centered B involves averaged fluxes for B update with certain stencil

15 CapSel DivB - 15 Diffusive treatment add diffusion type source terms to energy/induction equation diffuse B errors at maximal rate maximal rate = allowed by unchanged CFL condition for parabolic equation B t = η D 2 B get CFL constraint t < x 2 /η D take η D x 2 and use sources diffusion coefficient C d B C d x 2 ( B ) C d x 2 ( B )

16 CapSel DivB - 16 source terms destroy strict conservation (like Powell) could retain source term for induction equation alone parabolic approach: errors are damped at maximal rate hyperbolic divergence cleaning strategy: Dedner et al. (2002) B transported at maximal admissable speed AND damped simultaneously

17 CapSel DivB - 17 Numerical tests taken from Tóth (2000): 7 strategies on 9 2D MHD tests fair comparison: same (approximate Riemann solver) base scheme best strategy is projection/field-cd/flux-ct scheme need a strategy for B, usually does not matter too much which

18 CapSel DivB D rotated shock tube: 1D Riemann problem in 2D rotated over angle must maintain constant B (parallel to shock tube direction) RH violation by Powell source terms

19 CapSel DivB - 19a Orszag-Tang (1979) MHD vortex simulation (Picone & Dahlburg 1991) 2D domain [0, 2π] 2 with double periodic sides t = 0 uniform ρ = 25/9, p = 5/3, velocity vortex v = ( sin y, sin x) magnetic islands horizontal wavelength B = ( sin y, sin 2x) with γ = 5/3: Mach 1 flow conditions mimics evolution to compressible (supersonic) MHD turbulence

20 CapSel Orszag-Tang vortex problem, temperature at t = 3.14 DivB - 19b

21 CapSel DivB - 20 References G. Tóth, J. Comp. Phys. 161, 605 (2000) K.G. Powell et al., J. Comp. Phys. 154, 284 (1999) P.J. Dellar, J. Comp. Phys. 172, 392 (2001) P. Janhunen, J. Comp. Phys. 160, 649 (2000) G. Tóth, J. Comp. Phys. 182, 346 (2002) J.U. Brackbill, D.C. Barnes, J. Comp. Phys. 35, 426 (1980) A. Dedner et al., J. Comp. Phys. 175, 645 (2002) W. Dai, P.R. Woodward, Astrophysical Journal 494, 317 (1998) C.R. Evans, J.F. Hawley, Astrophysical Journal 332, 659 (1988)