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On Some Recent Developments in Numerical Methods for Relativistic MHDPowerPoint Presentation

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

Recent reviews in Living Reviews in Relativity Relativistic MHD

(www.livingreviews.org):

(i) Marti & Muller, 2003, “Numerical HD in Special Relativity

(ii) Font, 2003, “Numerical HD in General Relativity”;

Optimistic plan of the talk Relativistic MHD

- 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.

II. Relativistic MHDCONSERVATION LAWSAND HYPERBOLIC WAVES

- Single conservation law

U - conserved quantity,

F- flux of U,

S - source of U

- System of conservation laws

- 1D system of conservation laws with no source terms Relativistic MHD

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

- Continuous hyperbolic waves Relativistic MHD

- Jacobean matrix

Eigenvalue problem:

- wavespeed of k-th mode

- transported information

Fast, Slow, Alfven, and Entropy modes in MHD

- Shock waves Relativistic MHD

- shock equations

s – shock speed

- Hyperbolic shocks:
- As UraUl one has
- (i) (Ur -Ul) ar ;
- (ii) s alk.
- e.g. Fast, Slow, Alfven, and
- Entropy discontinuities in MHD

continuous hyperbolic wave

There exist other, non-hyperbolic shock

solutions !

hyperbolic shock

III. Relativistic MHDNON-CONSERVATIVEAND 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!

(b) Relativistic MHDConservative school

where

- 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

Komissarov (1999,2002,2004);

Anton et al.(2005).

- Linear Riemann Solver due to Roe (1980,1981) Relativistic MHD

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

(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?!

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:

- Kurganov-Tadmor (KT): Relativistic MHD

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

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?

IV. Relativistic MHDGOING HIGHER ORDERAND 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.

- The nature of numerical diffusion Relativistic MHD

second order scheme Relativistic MHD

first order scheme

- first order accurate Relativistic MHD - 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?

VI. Relativistic MHDKEEPINGB 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”

( 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:

conservation laws

- the “telegraph equation” for div B

VII. Relativistic MHDGOING 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.

- 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

VIII. “ Fiducial Observer (FIDO) STIFFNESS”OF MAGNETICALLY-DOMINATED MHD

Magnetohydrodynamics Magnetodynamics

This has 4 independent components.This has only 2 !

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”.

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

Thank you!

LRS HLL

LRS HLL

LRS HLL

1st order; no diffusion. 1st order; LLF-type diffusion

rarefaction shock

1st order

2nd order

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