Finite differencing of the transport equation

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Finite differencing of the transport equation

We split the description of the finite differencing of the Boltzmann equation (15) into separate terms according to the list in Eq. (16)-(22). Each term is prepared in its own subsection. In section we describe how all the individual terms are gathered to form one implicit solution of the complete transport equation and an operator split implicit solution of the hydrodynamics equations.

The knowledge of a complete set of primitive variables at a certain time allows the derivation of all other quantities of interest. Our choice of primitive variables is listed in table (). The discretization in space is indicated by the index $i$ . We adopt the convention that integer values of $i$ point to zone edges while $i+1/2$ refers to the mass center of the zone between edge $i$ and $i+1$ . We abbreviate half-valued indices with a prime, $i'\equiv i+1/2$ , in order to not further challenge the readability of the finite difference equations. Of order $100$ spatial zones are used on a dynamically adaptive grid. The particle distribution functions, $F_{i',j',k'}$ , additionally depend on the particle momentum phase space, i.e. the angle cosine of the particle propagation direction, $\mu$ , and the particle energy, $E$ , measured in the comoving frame. Both momentum phase space dimensions are discretized with zone center values, $\mu _{j'}$ and $E_{k'}$ , in compliance with the above-mentioned conventions. We use Gaussian quadrature in the angles $\mu _{j'}$ with weights $w_{j'}$ . They are derived from Legendre polynomials and normalized such that $\int _{-1}^{1}d\mu =\sum w_{j'}=2$ . The range from inwards to outwards propagation is currently resolved with $6$ different propagation angles, but we are free to choose a larger number of directions for higher resolution (see section ). The energy grid is set by geometrically increasing zone edge values $E_{k}$ . Following Bruenn_02, we define the zone center energies by

$\begin{displaymath} E_{k'}=\sqrt{\frac{E_{k+1}^{2}+E_{k+1}E_{k}+E_{k}^{2}}{3}}.\end{displaymath}$

This choice implements the phase space volume $E_{k'}^{2}dE_{k'}=\left( E_{k+1}^{3}-E_{k}^{3}\right) /3$ in an exact manner. The zone center energies also form a geometric series. We currently use $12$ zone center energies ranging from $3$ MeV to $300$ MeV, but this number can also be increased for higher resolution (see section

). Thermodynamical and compositional quantities are obtained from the primitive variables $\rho$ , $T$ , and $Y_{e}$ by the equation of state. Properties of the particle radiation field are obtained by the evaluation of the expectation value of the corresponding particle property with the primitive particle distribution function $F$ .

Table: The primitive variables are listed with an index $i$ if they live on zone edges and an index $i'\protect$ if they live on zone centers. The momentum phase space for the neutrino distribution function is labelled with an index $j'\protect$ for the angle cosine of the propagation direction and an index $k'\protect$ for the neutrino energy.

	primitive variable		primitive variable
$a_{i}$	enclosed rest mass	$\rho _{i'}$	rest mass density
$r_{i}$	radius	$T_{i'}$	temperature
$u_{i}$	velocity	$Y_{e,i'}$	electron fraction
$m_{i}$	enclosed gravitational mass	$\alpha _{i'}$	lapse function
		$F_{i',j',k'}$	neutrino distribution function

Subsections

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ApJS preprint doi:10.1086/380191