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. Recent reviews in Living Reviews in Relativity
On Some Recent Developments in Numerical Methods for Relativistic MHD
as seen by an astrophysicist
with some experience in computer simulations
School of Mathematics
University of Leeds
Recent reviews in Living Reviews in Relativity Relativistic MHD
(i) Marti & Muller, 2003, “Numerical HD in Special Relativity
(ii) Font, 2003, “Numerical HD in General Relativity”;
Optimistic plan of the talk Relativistic MHD
II. Relativistic MHDCONSERVATION LAWSAND HYPERBOLIC WAVES
U - conserved quantity,
F- flux of U,
S - source of U
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 !
Usually there exist auxiliary (primitive) variables, P, such that U and F are simple explicit functions of P.
- Jacobean matrix
- transported information
Fast, Slow, Alfven, and Entropy modes in MHD
- shock equations
s – shock speed
continuous hyperbolic wave
There exist other, non-hyperbolic shock
III. Relativistic MHDNON-CONSERVATIVEAND CONSERVATIVE SCHEMES
(a) Non-conservative school (orthodox)
(b) Relativistic MHDConservative school
- exchange by the same amount of U
between the neighbouring cells
IV. Relativistic MHDCAUSALAND 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
Anton et al.(2005).
at t = tn
Constant flux through
the interface x = xi+1/2 :
(b) Relativistic MHDNon-causal (central) fluxes
Why not to try something simpler, like
Well, this leads to instability.
Why not to dump it with indiscriminate diffusion?!
This leads to the following numerical flux
, where L is the highest wavespeed
on the grid. Very high diffusion!
where l(k) are the local wavespeeds (Local Lax flux)
This makes some use
Implemented in the Relativistic MHD schemes by: Relativistic MHD
* 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?
LRS – linear Riemann solver
(ii) Stationary tangent discontinuity Relativistic MHD
(iii) Stationary slow shock Relativistic MHD
(iv). Fast moving slow shock Relativistic MHD
IV. Relativistic MHDGOING HIGHER ORDERAND ADAPTIVE
second order scheme Relativistic MHD
first order scheme
V. Relativistic MHDGOING MULTI-DIMENSIONAL.
VI. Relativistic MHDKEEPINGB DIVERGENCE FREE.
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!
with charge density
( Relativistic MHDii) 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)
(iii) Relativistic MHDConstrained 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)
(iv) Relativistic MHD Diffusive cleaning
Integrate this modified induction
equation (not a conservation law )
- diffusion of div B
Implemented in Anninos et al (2005)
( Relativistic MHDv) Telegraph cleaning by Dedner et al.(2002)
Introduce new scalar variable, Y, additional evolution equation (for Y), and
modify the induction equation as follows:
VII. Relativistic MHDGOING GENERAL RELATIVISTIC
- covariant continuity equation
- continuity equation in partial derivatives
Utilization of central fluxes
- abg-representation of the
metric form. Vector b is
the grid velocity in FIDO’s
Here we have got a
Riemann problem with
Anton et al. (2005)
VIII. “ Fiducial Observer (FIDO) STIFFNESS”OF MAGNETICALLY-DOMINATED MHD
This has 4 independent components.This has only 2 !
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
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)
1st order; no diffusion. 1st order; LLF-type diffusion