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$\begin{figure} \par\psfrag{a}{\bf a} \psfrag{b}{\bf b} \includegraphics[width=7.55cm,clip]{3401fg1.eps} \end{figure}$ Figure 1: Comparing the long a) and short b) characteristics method. For the long characteristics method, the closer one gets to the source, the more rays pass through (approximately) the same part of a cell, resulting in a large number of redundant calculations. The short characteristics method does not suffer from this, since here column densities are interpolated from cells that have been dealt with previously, so only the contributions to the column density of the short ray sections that pass from cell to cell need to be computed.
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In the text

$\begin{figure} \par\psfrag{processor 0}{\bf processor 0} \psfrag{processor 1}{\bf processor 1} \includegraphics[width=7.65cm,clip]{3401fg2.eps} \end{figure}$ Figure 2: Two-dimensional example of an AMR hierarchy distributed over two different processors. Here, each patch contains 4 $\times$ 4 cells.
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In the text

$\begin{figure} \par\psfrag{a}{\bf a} \psfrag{b}{\bf b} \includegraphics[width=7.65cm,clip]{3401fg3.eps} \end{figure}$ Figure 3: Two-dimensional example of ray sections for a single patch. Local contributions to the column density are indicated by ray sections that terminate at cell centres a), whereas contributions that are to be communicated between processors, and are subsequently used in an interpolation step, terminate at cell corners b). The source lies outside of the patch in the direction of the lower left corner.
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In the text

$\begin{figure} \par\psfrag{processor 0}{\bf processor 0} \psfrag{processor 1}{\b... ...5} \psfrag{16}{16} \includegraphics[width=7.65cm,clip]{3401fg4.eps} \end{figure}$ Figure 4: Two-dimensional example of a "patch-mapping'' for a computational domain that is split over two different processors. In the top row the local ids of the patches on the different processors are shown. The mapping of these patch ids onto the patch-mapping array is shown in the middle row. The bottom row shows the global patch-mapping after the local patch-mappings have been communicated. Tracing the depicted ray results in the patch list $\{1,4,10,13,16\}$ . The patch-mapping entries visited during the ray tracing are shown in grey.
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In the text

$\begin{figure} \par\psfrag{l1}{$l_1$ } \psfrag{l2}{$l_2$ } \psfrag{le}{$l_e$ } \... ... } \psfrag{e}{$e$ } \includegraphics[width=5.2cm,clip]{3401fg5.eps} \end{figure}$ Figure 5: Two-dimensional illustration of the linear interpolation scheme used to accumulate local column density contributions. Shown are the ray sections used in the interpolation (see text for further details).
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In the text

$\begin{figure} \par\psfrag{e}{$e$ } \psfrag{1}{$1$ } \psfrag{2}{$2$ } \psfrag{3}... ... } \psfrag{8}{$8$ } \includegraphics[width=7.8cm,clip]{3401fg6.eps} \end{figure}$ Figure 6: Illustration of the interpolation scheme in three dimensions. For clarity we show outlines of cells on patch faces only. In the top image we show a ray r that exits the patch at location e through a cell face, together with the ray sections used in the interpolation that terminate at the corners of this cell face. The image at the bottom shows the cell face in more detail, where we indicated the cell corners by 1, 2, 3, and 4. In addition to these cell corners, ray sections used in the interpolation that terminate at 5, 6, 7, and 8 are also indicated (see text for further details).
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In the text

$\begin{figure} \par\includegraphics[width=6.7cm,clip]{3401fg7.eps} \end{figure}$ Figure 7: Summary of steps taken in the hybrid characteristics method. In the top image we show ray sections that represent local contributions to the column density (summary step 3), whereas ray sections that represent values of column density that need to be communicated are shown in the centre image (summary step 4). Note that only those values on patch faces located farthest away from the source need to be communicated. In the bottom image we show an example of the interpolation of these local values for a particular destination cell (summary step 6). Note that there is no need to interpolate the value for the final ray section in the destination patch since its value was already calculated previously (summary step 3).
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In the text

$\begin{figure} \par\includegraphics[width=6.5cm,clip]{3401fg8a.eps}\par\vspace*{1.0cm} \includegraphics[width=6.5cm,clip]{3401fg8b.eps} \end{figure}$ Figure 8: Test 1: values of HI column density for the case of a single point source in a two-dimensional domain with a 1/r² density distribution. Shown are one-dimensional cuts along the y-direction through the source located at the centre of the domain ( top two panels) and at 3/4 of the domain ( bottom two panels). In the panels at the first and third row, the solid line indicates the result for the long, whereas the crosses indicate the result for the hybrid characteristics method. In the panels at the second and fourth row, the ratio (hybrid/long) of HI column density values is shown.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{3401fg9a.eps}\par\includeg... ...]{3401fg9c.eps}\par\includegraphics[width=7.5cm,clip]{3401fg9d.eps} \end{figure}$ Figure 9: Test 2: values of $\log_{10}$ of the column density ( top two panels) and $\log_{10}$ of the HI ionization fraction ( bottom two panels) for the case of a single point source in an environment with a homogeneous density distribution containing neutral clumps with higher density. The source is located at (0.0,0.5,0.5) $\times$ $10^{18}~{\rm cm}$ , and the clumps are located at (1.0,0.3,0.5) $\times$ $10^{18}~{\rm cm}$ and (1.0,0.7,0.5) $\times$ $10^{18}~{\rm cm}$ , respectively. Shown are color coded plots of xy-cuts through the centre of the domain, at z=0.5 $\times$ $10^{18}~{\rm cm}$ ( top and third row) and xz-cuts through the centre of the bottom clump, at y=0.3 $\times$ $10^{18}~{\rm cm}$ ( second and bottom row). As an example, the image in the second row shows the AMR patch distribution superimposed, where each patch contains 16³ cells.
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In the text

$\begin{figure} \par\includegraphics[width=7.5cm,clip]{3401f10a.eps}\par\includeg... ...]{3401f10b.eps}\par\includegraphics[width=7.5cm,clip]{3401f10c.eps} \end{figure}$ Figure 10: Test 3: two sources in an environment with a homogeneous density distribution containing a neutral clump with higher density. The clump is located at (1,1,1) $\times$ $10^{18}~{\rm cm}$ , and the sources are located at (0,1,1) $\times$ $10^{18}~{\rm cm}$ and (0.5,0,1) $\times$ $10^{18}~{\rm cm}$ , respectively. Shown are color coded plots of $\log_{10}$ of the HI column density ( top, middle) for the two different sources, and $\log_{10}$ of the HI ionization fraction ( bottom). All plots show xy-cuts at $z=10^{18}~{\rm cm}$ through the centre of the domain.
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In the text

$\begin{figure} \par\includegraphics[height=3.5cm,width=8.3cm,clip]{3401f11a.eps}... ...5cm} \includegraphics[height=3.5cm,width=8.3cm,clip]{3401f11l.eps} \end{figure}$ Figure 11: Test 4: snapshots of the evolution of the $\log_{10}$ of the mass density ( left) and the $\log_{10}$ of the HI ionization fraction ( right) for the case of a single point source in an environment with a homogeneous density distribution containing two neutral clumps with higher density. The source is located at (0.0,0.5,0.5) $\times$ $10^{18}~{\rm cm}$ , and the clumps are located at (1.0,0.3,0.5) $\times$ $10^{18}~{\rm cm}$ and (1.0,0.7,0.5) $\times$ $10^{18}~{\rm cm}$ , respectively. Shown are color coded plots of xy-cuts through the centre of the domain (z=0.5 $\times$ $10^{18}~{\rm cm}$ ) at times $t=0~{\rm yr}$ ( first row), $t=792~{\rm yr}$ ( second row), $t=1584~{\rm yr}$ ( third row), $t=2377~{\rm yr}$ ( fourth row), $t=3169~{\rm yr}$ ( fifth row), and $t=3961~{\rm yr}$ ( sixth row).
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{3401fg12.eps} \end{figure}$ Figure 12: Performance of the main components of a radiation hydrodynamics calculation. Plotted are the total calculation time in seconds for the hydrodynamics (+), AMR ( $\times$ ), and radiation (*) parts of the algorithm as a function of number of processors.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{3401fg13.eps} \end{figure}$ Figure 13: Performance of the different steps in the radiation part of the hybrid characteristics method. Plotted are the total calculation time in seconds for the local ray trace (+), communication ( $\times$ ), accumulate/interpolate (*), and ionization ( $\square$ ) parts of the calculation as a function of the number of processors. For this specific test, the communication takes as much time as the rest of the calculation when using 64 processors. As is explained in the text, for patches with a larger number of cells, this constraint may become less severe.
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In the text

$\begin{figure} \par\psfrag{txi}{$t_x^{\rm i}$ } \psfrag{tyi}{$t_y^{\rm i}$ } \ps... ...g{Dy}{$\Delta y$ } \includegraphics[width=5.8cm,clip]{3401fgA1.eps} \end{figure}$ Figure A.1: Explanation of different quantities used in the fast voxel traversal method. Shown is a single patch with a local long characteristic ray section.

In the text

$\begin{figure} \par\psfrag{1}{$1$ } \psfrag{2}{$2$ } \psfrag{3}{$3$ } \psfrag{4}{$4$ } \includegraphics[width=7.55cm,clip]{3401fgB1.eps} \end{figure}$ Figure B.1: Two examples of short characteristics sweeping sequences for a single patch that may occur in practice. For the illustration on the left, the source is located inside the patch at the starting point of curve 1. In this case the four space filling curves shown should be swept in the order indicated. For the patch on the right, the source is external to the patch, and lies in the direction of the lower left corner, so there is only one curve that needs to be swept.

In the text

$\begin{figure} \par\psfrag{c}{$c$ } \psfrag{c1}{$c_1$ } \psfrag{c2}{$c_2$ } \psf... ...psfrag{d2}{$d_2$ } \includegraphics[width=6.5cm,clip]{3401fgB2.eps} \end{figure}$ Figure B.2: Illustration of the interpolation scheme for a single cell used in the short characteristics method for the 2D ( left) and 3D ( right) case.

In the text

$\begin{figure} \par\psfrag{processor 0}{\bf processor 0} \psfrag{processor 1}{\bf processor 1} \includegraphics[width=7.65cm,clip]{3401fg2.eps} \end{figure}$	Figure 2: Two-dimensional example of an AMR hierarchy distributed over two different processors. Here, each patch contains 4 $\times$ 4 cells.
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$\begin{figure} \par\psfrag{l1}{$l_1$ } \psfrag{l2}{$l_2$ } \psfrag{le}{$l_e$ } \... ... } \psfrag{e}{$e$ } \includegraphics[width=5.2cm,clip]{3401fg5.eps} \end{figure}$	Figure 5: Two-dimensional illustration of the linear interpolation scheme used to accumulate local column density contributions. Shown are the ray sections used in the interpolation (see text for further details).
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$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{3401fg12.eps} \end{figure}$	Figure 12: Performance of the main components of a radiation hydrodynamics calculation. Plotted are the total calculation time in seconds for the hydrodynamics (+), AMR ( $\times$ ), and radiation (*) parts of the algorithm as a function of number of processors.
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