the $i$-th mesh is calculated by using adjacent 26 meshes, i.e. 27 meshes in total, with a Gaussian kernel of the smoothing length $\Delta L$. Doing this drastically reduces the number of iterations required to find the solutions. [ This smoothing may introduce the smearing of the I-fronts especially when the spacial resolution is poor. While we do not find such an effect in our test simulations as we will show later, this can be avoided by applying the smoothing only to optically thick meshes as done by @RSPH. ]{}

Test simulations {#sec:tests}
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In this section, we describe the tests we perform. In order to evaluate the accuracy of our tree-based RT algorithms, problems should involve many sources. Therefore some of the tests presented are neither simplest nor cleanest. All test problems are solved in three dimensions, with $128^3$ meshes unless otherwise stated.

Test 1 – Pure hydrogen isothermal ${\mbox{{H$\,${\sc ii}}}}$ region expansion {#stroemgren}
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![Test 1 – Images of the ${\ensuremath{\mbox{{H$\,${\sc i}}}}}$ fraction, cut through at the mid plane of the simulation box at $t = 500~{\rm Myr}$.[]{data-label="fig:strmap"}](fig4.eps){width="8.0cm"}

The first test is the classical problem of a ${\mbox{{H$\,${\sc ii}}}}$ region expansion in a static, homogeneous, and isothermal gas, 