On Some Recent Developments in Numerical Methods for Relativistic MHD

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On Some Recent Developments in Numerical
Methods for Relativistic MHD as seen by an
astrophysicist with some experience in computer
simulations Serguei Komissarov School of
Mathematics University of Leeds UK
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Recent reviews in Living Reviews in
Relativity (www.livingreviews.org)
(i) Marti Muller, 2003, Numerical HD in
Special Relativity (ii) Font, 2003, Numerical
HD in General Relativity

• 
  

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Optimistic plan of the talk

• Conservation laws and hyperbolic waves.
• Non-conservative (orthodox) and conservative
  (main stream) schools.
• Causal and central numerical fluxes in
  conservative schemes.
• Going higher order and adaptive.
• Going multi-dimensional.
• Keeping B divergence free.
• 7. Going General Relativistic.
• 8. Stiffness of magnetically-dominated MHD.
• 9. Intermediate (trans-Alfvenic) shocks.

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II. CONSERVATION LAWS AND HYPERBOLIC WAVES

• Single conservation law

U - conserved quantity, F- flux of U, S -
source of U

• System of conservation laws

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• 1D system of conservation laws with no source
  terms

In many cases F is known as only an implicit
function of U, f(U,F)0 . In relativistic MHD
the conversion of U into F involves solving a
system of complex nonlinear algebraic equations
numerically computationally expensive !

• Non-conservative form of conservation laws

Usually there exist auxiliary (primitive)
variables, P, such that U and F are simple
explicit functions of P.
where
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• Continuous hyperbolic waves

- Jacobean matrix
Eigenvalue problem

• wavespeed of k-th mode

- transported information
Fast, Slow, Alfven, and Entropy modes in MHD
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• Shock waves

- shock equations
s shock speed

• Hyperbolic shocks
• As Ur aUl one has
• (i) (Ur -Ul) a r
• (ii) s a lk.
• e.g. Fast, Slow, Alfven, and
• Entropy discontinuities in MHD

continuous hyperbolic wave
There exist other, non-hyperbolic shock
solutions !
hyperbolic shock
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III. NON-CONSERVATIVE AND CONSERVATIVE SCHEMES
(a) Non-conservative school (orthodox)

• Wilson (1972) a De Villiers Hawley
  (2003), Anninos et al.(2005)

• Finite-difference version of

• Artificial viscosity (physically motivated
  dissipation) is utilised to construct
• stable schemes
• (i) poor representation of shocks
• (ii) only low Lorentz factors (glt3)
• Why Ultra-Relativistic Numerical
  Hydrodynamics is Difficult
• 
  by Norman Winkler(1986)
• Anninos et al.,(2005) Go conservative!

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(b) Conservative school

where
- exchange by the same amount of U between
the neighbouring cells
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IV. CAUSAL AND CENTRAL NUMERICAL FLUXES
(a) Causal (upwind) fluxes
Utilize exact or approximate solutions for the
evolution of the initial discontinuity at the
cells interfaces (Riemann problems) to evaluate
fluxes.
Initial discontinuity
Its resolution
Implemented in the Relativistic MHD schemes by
Komissarov (1999,2002,2004)
Anton et al.(2005).
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• Linear Riemann Solver due to Roe (1980,1981)

linearization
at t tn
Riemann problem
Wave strengths

• a system of linear equations
• for the wave strengths, a(k)

Constant flux through the interface x xi1/2

transported information
wave strengths
wave speeds
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(b) Non-causal (central) fluxes
?
Why not to try something simpler, like Well,
this leads to instability. Why not to dump it
with indiscriminate diffusion?! The modified
equation
where
artificial diffusion
This leads to the following numerical flux
, where L is the highest wavespeed on the
grid. Very high diffusion!

• Lax

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• Kurganov-Tadmor (KT)

where l(k) are the local wavespeeds (Local Lax
flux)

• Harten, Lax van Leer (HLL)

artificial diffusion
where
This makes some use of causality
i1
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Implemented in the Relativistic MHD schemes by
HLL Del Zanna Bucciantini (2003), Gammie
et al. (2003) Duez et
al.(2005), Anton et al. (2005). KT
Anton et al. (2005), Anninos et al.(2005)
Koide
et al.(1996,1999)
The central schemes are claimed to be as good as
the causal ones !
Are they really?
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• 1D test simulations
• (i) Stationary fast shock

LRS linear Riemann solver
HLL
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(ii) Stationary tangent discontinuity
LRS
HLL
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(iii) Stationary slow shock
LRS
HLL
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(iv). Fast moving slow shock
LRS
HLL
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IV. GOING HIGHER ORDER AND ADAPTIVE

• Fully causal fluxes provide better numerical
  representation of stationary
• and slow moving shocks/discontinuities ( see
  also Mignone Bodo, 2005)
• However for fast moving moving
  shocks/discontinuities they give similar
• results to central fluxes. (Lucas-Serrano et
  al. 2004, Anton et al. 2005).
• How to improve the
  representation of shocks moving rapidly
• 
  across the grid ?

• Use adaptive grids to increase resolution near
  shocks.
• Falle Komissarov (1996), Anninos et
  al. (2005)
• (ii) Use sub-cell resolution to reduce
  numerical diffusion.

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• The nature of numerical diffusion

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second order scheme
first order scheme
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• first order accurate - piece-wise constant
  reconstruction
• second order accurate - piece-wise linear
  reconstruction
• Komissarov (1999,2002,2004), Gammie et
  al.(2003),
• Anton et al. (2005).
• third order accurate - piece-wise parabolic
  reconstruction
• Del Zanna Bucciantini (2003), Duez et al.
  (2005).
• In astrophysical simulations Del Zanna
  Buucciantini are
• forced to reduce their scheme to second
  order (oscillations at shocks) !?
• THERE IS THE OPTIMUM?

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V. GOING MULTI-DIMENSIONAL.
F
Ui,j
F
F
F
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VI. KEEPING B DIVERGENCE FREE.

• Differential equations

• the evolution equation can
• keep B divergence free !

• Difference equations may not have such a nice
  property.

What do we do about this ?
(i) Absolutely nothing. Treat the induction
equation as all other conservation laws
( Koide et al. 1996,1999).
Such schemes crash all too often!
magnetic monopoles with charge density
-magnetostatic force
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(ii) Toths constrained transport.
Use the modified flux F that is such a linear
combination of normal fluxes at neighbouring
interfaces that the corner- -centred numerical
representation of divB is kept invariant during
integration.
Implemented in Gammie et al.(2003), and
Duez et al.(2005)
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(iii) Constrained Transport of Evans Hawley.
Use staggered grid (with B defined at the cell
interfaces) and evolve magnetic fluxes through
the cell interfaces using the electric field
evaluated at the cell edges. This keeps the
following cell-centred numerical
representation of divB invariant
Implemented in Komissarov (1999,2002,2004), de
Villiers Hawley (2003),
Del Zanna et al.(2003), and Anton et
al.(2005)
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(iv) Diffusive cleaning
Integrate this modified induction equation (not
a conservation law )
- diffusion of div B
Implemented in Anninos et al (2005)
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(v) Telegraph cleaning by Dedner et al.(2002)
Introduce new scalar variable, Y, additional
evolution equation (for Y), and modify the
induction equation as follows
conservation laws

• the telegraph equation for div B

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VII. GOING GENERAL RELATIVISTIC

• GRMHD equations can also be written as
  conservation laws

- covariant continuity equation
- continuity equation in partial derivatives

• determinant of the
• metric tensor

• conservative form of the continuity equation.
• tx0/c and tconst defines a space-like
• hyper-surface of space-time (absolute space)

U
F i
Utilization of central fluxes is
straightforward.
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• Riemann problems can be solved in the frame of
  the local Fiducial Observer (FIDO)
• using Special Relativistic Riemann solvers. A
  FIDO is at rest in the absolute space
• but generally is not at rest relative to the
  coordinate grid (Papadopoulos Font 1998)

- abg-representation of the metric form.
Vector b is the grid velocity in FIDOs
frame
Here we have got a Riemann problem with moving
interface
Implemented in Komissarov (2001,2004), Anton et
al. (2005)
FIDOs frame
coordinate grid
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VIII. STIFFNESS OF MAGNETICALLY-DOMINATED MHD
Magnetohydrodynamics
Magnetodynamics
This has 4 independent components.
This has only 2 !
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What to do if such magnetically-dominated regions
do develop ?

• Solve the equations of Magnetodynamics (e.g.
  Komissarov 2001,2004)
• Pump new plasma in order to avoid running
  into the danger zone.

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VIII. INTERMEDIATE SHOCKS
This numerical solution of the relativistic Brio
Wu test problem is corrupted by the presence
of non-physical compound wave which involves a
non-evolutionary intermediate shock. Such
shocks are known to pop up in non-relativistic
MHD simulations. Brio Wu (1988) Falle
Komissarov (2001) de Sterk Poedts
(2001) Torrilhon Balsara (2004) Almost
nothing is known about the relativistic
intermediate shocks. How to avoid them? Use
very high resolution. Torrilhon Balsara
(2004)
fast rarefaction
compound wave
slow rarefaction
fast rarefaction
intermediate shock
slow shock
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Yes! This is finally over ! Thank you!
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Stationary Contact LRS
HLL
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Slowly Moving Contact LRS
HLL
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Stationary Current Sheet LRS
HLL
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Fast Rarefaction Wave (LRS) 1st
order no diffusion.
1st order LLF-type diffusion

rarefaction shock
1st order
2nd order