1. 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

2. (ii) Font, 2003, “Numerical HD in General Relativity”;
Recent reviews in Living Reviews in Relativity (
(i) Marti & Muller, 2003, “Numerical HD in Special Relativity
(ii) Font, 2003, “Numerical HD in General Relativity”;

3. 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.

4. II. CONSERVATION LAWS AND HYPERBOLIC WAVES
Single conservation law
U - conserved quantity,
F- flux of U,
S - source of U
System of conservation laws

5. 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

6. - transported information
Continuous hyperbolic waves
- Jacobean matrix
Eigenvalue problem:
wavespeed of k-th mode
- transported information
Fast, Slow, Alfven, and Entropy modes in MHD

7. Entropy discontinuities in MHD
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

8. 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 (g<3);
“Why Ultra-Relativistic Numerical Hydrodynamics is Difficult”
by Norman & Winkler(1986);
Anninos et al.,(2005) : Go conservative!

9. (b) Conservative school
where
- exchange by the same amount of U
between the neighbouring cells

10. 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).

11. 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 = xi+1/2 :
transported
information
wave strengths
wave speeds

12. (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
This leads to the following numerical flux
artificial
diffusion
, where L is the highest wavespeed
on the grid. Very high diffusion!
Lax:

13. where l(k) are the local wavespeeds (Local Lax flux)
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:
i+1

14. 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?

15. (i) Stationary fast shock
1D test simulations
(i) Stationary fast shock
LRS – linear Riemann solver
HLL

16. (ii) Stationary tangent discontinuity
LRS
HLL

17. (iii) Stationary slow shock
LRS
HLL

18. (iv). Fast moving slow shock
LRS
HLL

19. 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.

20. The nature of numerical diffusion

21. second order scheme
first order scheme

22. 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?

23. V. GOING MULTI-DIMENSIONAL.F
Ui,j F F F

24. 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”

25. (ii) Toth’s 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)

26. Use staggered grid (with B defined at the cell
(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)

27. (iv) Diffusive cleaning
Integrate this modified induction
equation (not a conservation law )
- diffusion of div B
Implemented in Anninos et al (2005)

28. (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

29. 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.
t=x0/c and t=const defines a space-like
hyper-surface of space-time (absolute space) U
F i
Utilization of central fluxes
is straightforward.

30. 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 FIDO’s
frame
Here we have got a
Riemann problem with
moving interface
Implemented in
Komissarov (2001,2004),
Anton et al. (2005)
FIDO’s frame
coordinate grid

31. VIII. “STIFFNESS” OF MAGNETICALLY-DOMINATED MHD
Magnetohydrodynamics Magnetodynamics
This has 4 independent components This has only 2 !

32. 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”.

33. 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
intermediate
shock
fast
rarefaction
slow shock

34. Yes! This is finally over !
Thank you!

35. Stationary Contact
LRS HLL

36. Slowly Moving Contact
LRS HLL

37. Stationary Current Sheet
LRS HLL

38. Fast Rarefaction Wave (LRS)
1st order; no diffusion st order; LLF-type diffusion
rarefaction shock
1st order
2nd order