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0001 \documentclass{article}
0002
0003 \usepackage{amsmath}
0004
0005 \newcommand{\dd}{\partial}
0006
0007 \title{Technical notes}
0008 \author{Iurii Karpenko}
0009
0010 \begin{document}
0011 \maketitle
0012
0013 \section{Ideal-viscous splitting}
0014
0015 Here I explain how an updated method of ideal-viscous splitting is implemented in \texttt{vHLLE}. To derive it, one starts from Eqs. (11) in \cite{Karpenko:2013wva}:
0016
0017 \begin{align}\label{eqns1}
0018 &\tilde\dd_\nu(\tau \tilde T^{\tau\nu})+\frac{1}{\tau}(\tau \tilde T^{\eta\eta})=0, \nonumber\\
0019 &\tilde\dd_\nu (\tau \tilde T^{x\nu}) = 0, \nonumber\\
0020 &\tilde\dd_\nu (\tau \tilde T^{y\nu}) = 0,\\
0021 &\tilde\dd_\nu(\tau \tilde T^{\eta\nu}) + \frac{1}{\tau} (\tau \tilde T^{\eta\tau}) = 0, \nonumber\\
0022 &\tilde\dd_\nu(\tau \tilde N_c^\nu)=0. \nonumber
0023 \end{align}
0024 The numerical finite volume representation of the emergy-momentum conservation part is:
0025 \begin{align}\label{eq-num1}
0026 &\frac{1}{\Delta\tau}\left[ (\tau+\Delta\tau)(Q^\mu_{id,n+1}+\delta Q^\mu_{n+1}) - \tau (Q^\mu_{id,n}+\delta Q^\mu_n) \right] \nonumber\\
0027 &+\sum_{\alpha=1...3}\frac{(\tau+\Delta\tau/2)}{\Delta x_\alpha}\left[ F^{\mu\alpha}_{id,i+1/2}+\delta F^{\mu\alpha}_{i+1/2} - F^{\mu\alpha}_{id,i-1/2}-\delta F^{\mu\alpha}_{i-1/2} \right] \nonumber\\
0028 &=(\tau+\Delta\tau/2)(S^\mu_{n+1/2}+\delta S^\mu_{n+1/2}).
0029 \end{align}
0030 where second order accurate method is assumed (therefore half-step and cell-edge values), and $\delta Q$ and $\delta F$ denote viscous corrections to conserved variables and fluxes, respectively.\\
0031 The terms in Eqs.~\ref{eq-num1} can be rearranged as follows:
0032 \begin{align}
0033 &\left[ (\tau+\Delta\tau)Q^\mu_{id,n+1} - \tau Q^\mu_{id,n} \right]
0034 +\sum_{\alpha}\frac{(\tau+\Delta\tau/2)}{\Delta x_\alpha}\left[ F^{\mu\alpha}_{id,i+1/2}+\delta F^{\mu\alpha}_{i+1/2} - F^{\mu\alpha}_{id,i-1/2}-\delta F^{\mu\alpha}_{i-1/2} \right] \nonumber\\
0035 &=(\tau+\Delta\tau/2)(S^\mu_{n+1/2}+\delta S^\mu_{n+1/2}) + (\tau\delta Q^\mu_n - (\tau+\Delta\tau)\delta Q^\mu_{n+1})
0036 \end{align}
0037 The abouve form of equations mean that that in basic hydrodynamic (i.e. energy-momentum conservation) equations one can follow the evolution of ideal part of the conserved variables $Q_{id}^\mu$ only, when extra source terms $(\tau\delta Q^\mu_n - (\tau+\Delta\tau)\delta Q^\mu_{n+1})$ are included in their numerical evolution equations.
0038
0039 \begin{thebibliography}{0}
0040 \bibitem{Karpenko:2013wva}
0041 I.~Karpenko, P.~Huovinen and M.~Bleicher,
0042 %``A 3+1 dimensional viscous hydrodynamic code for relativistic heavy ion collisions,''
0043 Comput.\ Phys.\ Commun.\ {\bf 185} (2014) 3016
0044 %[arXiv:1312.4160 [nucl-th]].
0045 \end{thebibliography}
0046
0047 \end{document}
