StarBench: The D-type expansion of an HII region
  StarBench is a project focused on benchmarking and validating different
star-formation and stellar feedback codes. In this first StarBench paper we
perform a comparison study of the D-type expansion of an HII region. The aim of
this work is to understand the differences observed between the twelve
participating numerical codes against the various analytical expressions
examining the D-type phase of HII region expansion. To do this, we propose two
well-defined tests which are tackled by 1D and 3D grid- and SPH- based codes.
The first test examines the `early phase' D-type scenario during which the
mechanical pressure driving the expansion is significantly larger than the
thermal pressure of the neutral medium. The second test examines the `late
phase' D-type scenario during which the system relaxes to pressure equilibrium
with the external medium. Although they are mutually in excellent agreement,
all twelve participating codes follow a modified expansion law that deviates
significantly from the classical Spitzer solution in both scenarios. We present
a semi-empirical formula combining the two different solutions appropriate to
both early and late phases that agrees with high-resolution simulations to
$\lesssim2\%$. This formula provides a much better benchmark solution for code
validation than the Spitzer solution. The present comparison has validated the
participating codes and through this project we provide a dataset for
calibrating the treatment of ionizing radiation hydrodynamics codes.
