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 MathematicsUniversity of Leeds

UK

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”;

Optimistic plan of the talk

1. Conservation laws and hyperbolic waves.2. Non-conservative (orthodox) and conservative (main stream) schools.3. Causal and central numerical fluxes in conservative schemes.4. Going higher order and adaptive.5. Going multi-dimensional.6. Keeping B divergence free. 7. Going General Relativistic. 8. Stiffness of magnetically-dominated MHD.9. Intermediate (trans-Alfvenic) shocks.

II. CONSERVATION LAWS AND HYPERBOLIC WAVES

• Single conservation lawU - conserved quantity, F- flux of U, S - source of U

• System of conservation laws

• 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 !

where

• Non-conservative form of conservation laws

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

• Continuous hyperbolic waves

- Jacobean matrix

Eigenvalue problem:

- wavespeed of k-th mode - transported information

Fast, Slow, Alfven, and Entropy modes in MHD

• Shock waves

- shock equations

s – shock speed

continuous hyperbolic wave

hyperbolic shock

• Hyperbolic shocks:

As Ur Ul one has (i) (Ur -Ul) r ; (ii) s k.

e.g. Fast, Slow, Alfven, and Entropy discontinuities in MHD

There exist other, non-hyperbolic shock solutions !

III. NON-CONSERVATIVE AND CONSERVATIVE SCHEMES

(a) Non-conservative school (orthodox)

• Artificial viscosity (physically motivated dissipation) is utilised to construct stable schemes; • (i) poor representation of shocks; (ii) only low Lorentz factors “Why Ultra-Relativistic Numerical Hydrodynamics is Difficult” by Norman & Winkler(1986);

• Anninos et al.,(2005) : Go conservative!

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

• Finite-difference version of

(b) Conservative school

- exchange by the same amount of U between the neighbouring cells

where

IV. CAUSAL AND CENTRAL NUMERICAL FLUXES

(a) Causal (upwind) fluxes

Initial discontinuity Its resolution

Utilize exact or approximate solutions for the evolution of the initial discontinuity at the cells interfaces (Riemann problems) to evaluate fluxes.

Implemented in the Relativistic MHD schemes by Komissarov (1999,2002,2004); Anton et al.(2005).

• Linear Riemann Solver due to Roe (1980,1981)

linearization

Riemann problem:

Wave strengths:

Constant flux through the interface x = xi+1/2 :

transported information

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

wave strengthswave speeds

at t = tn

(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

• Lax: , where is the highest wavespeed on the grid. Very high diffusion!

• Kurganov-Tadmor (KT):

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

• Harten, Lax & van Leer (HLL) :

where

artificial diffusion

This makes some use of causality:

i+1

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?

• 1D test simulations (i) Stationary fast shock

LRS – linear Riemann solver HLL

(ii) Stationary tangent discontinuity

LRS HLL

(iii) Stationary slow shock

LRS HLL

(iv). Fast moving slow shock

LRS HLL

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 ?

(i) 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

first order scheme second order scheme

• 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?

V. GOING MULTI-DIMENSIONAL.

Ui,j

F

F

F

F

VI. KEEPING B DIVERGENCE FREE.

- the evolution equation can keep B divergence free !

• Differential equations

• 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

magnetic monopoleswith charge density -“magnetostatic force”

( Koide et al. 1996,1999).

Such schemes crash all too often!

(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)

(iii) Constrained Transport of Evans & Hawley.

Implemented in Komissarov (1999,2002,2004), de Villiers & Hawley (2003), Del Zanna et al.(2003), and Anton et al.(2005)

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

(iv) Diffusive cleaning

Integrate this modified induction equation (not a conservation law )

- diffusion of div B

Implemented in Anninos et al (2005)

(v) Telegraph cleaning by Dedner et al.(2002)

Introduce new scalar variable, , additional evolution equation (for , and modify the induction equation as follows:

-the “telegraph equation” for div B

conservation laws

VII. GOING GENERAL RELATIVISTIC

• GRMHD equations can also be written as conservation laws;

- covariant continuity equation

- continuity equation in partial derivatives

- conservative form of the continuity equation. t=x0/c and t=const defines a space-like hyper-surface of space-time (absolute space)

-determinant of the metric tensor

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)

--representation of the metric form. Vector is the grid velocity in FIDO’s frame

Here we have got aRiemann problem with moving interface

coordinate gridFIDO’s frame

Implemented inKomissarov (2001,2004),Anton et al. (2005)

VIII. “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”.

compound wave

fast rarefaction

fast rarefaction

slow shock

intermediateshock

slow rarefaction

VIII. INTERMEDIATE SHOCKSThis 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)

Yes! This is finally over !

Thank you!

Stationary Contact

LRS HLL

Slowly Moving Contact

LRS HLL

Stationary Current Sheet

LRS HLL

Fast Rarefaction Wave (LRS)

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

rarefaction shock 1st order

2nd order