Shear stress tensor profiles at specific time

Shear stress tensor profile along x-axis at $\tau$=1.6fm/c (HS+UH, VISH2+1 with full I-S eqns.):

The curves are labelled by the grid sizes used in the numerical calculation. For example, $0.1-0.04$ means $\Delta x=\Delta y=0.1$ fm, $\Delta \tau =0.04$ fm/c.

This figure was updated by H.S. on 09/08/2008 after correcting a small coding error in VISH2+1; it now uses totally identical initializations for the H&S and R&R equations. The old figure and discussion (posted by H.S. on 08/04/2008) can be found at Note for Shear stress tensor profiles at specific time.

Shown is a comparison between the following different versions of the "full I-S" eqns.:

• full I-S eqn-(R&R): $\Delta _{j}^{m}\Delta _{k}^{n}{\dot {\pi }}^{jk}=-{\frac {1}{\tau _{\pi }}}(\pi ^{mn}{-}2\eta \sigma ^{mn})+{\frac {1}{2}}\pi ^{mn}\left(5D(\ln T)-\partial _{k}u^{k}\right)$
  
• full I-S eqn-(S&H):$\Delta _{j}^{m}\Delta _{k}^{n}{\dot {\pi }}^{jk}=-{\frac {1}{\tau _{\pi }}}(\pi ^{mn}{-}2\eta \sigma ^{mn})-{\frac {1}{2}}\pi ^{mn}{\frac {\eta T}{\tau _{\pi }}}d_{k}\left({\frac {\tau _{\pi }}{\eta T}}u^{k}\right)$
  

For an ideal gas of massless gluons (as used here) these two equations are mathematically identical. Any differences are thus due to coding inefficiencies or rounding errors. The coincidence of the black and orange lines shows that, after correcting the previous coding error, the S&H and R&R forms of the equations give identical results, as they should. In this figure, we also investigate the effects of rounding errors, by showing two different choices for evaluating the last term in S&H eqn. The red lines are for directly using ${\frac {\eta T}{\tau _{\pi }}}d_{k}\left({\frac {\tau _{\pi }}{\eta T}}u^{k}\right)$; this develops large rounding errors near the edge of the fireball, due to a very small value of ${\frac {\eta T}{\tau _{\pi }}}$. This problem is fixed by changing ${\frac {\eta T}{\tau _{\pi }}}d_{k}\left({\frac {\tau _{\pi }}{\eta T}}u^{k}\right)$ to $(d_{k}u^{k}+Dln\left({\frac {\tau _{\pi }}{\eta T}}\right))$, shown by the dashed orange lines for the same grid spacing. One also sees that for $\Delta x=\Delta y=0.1$ fm, $\Delta \tau =0.04$ fm/c (orange and black) we still have small discretization errors, which disappear when the grid spacing is reduced by another factor 2 (purple and green).

Other formats and data tables for this figure can be downloaded here: Au-global-0508.tar

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