  Modelling galaxy formation in hydrodynamic simulations has increasingly
adopted various radiative transfer methods to account for photoionization
feedback from young massive stars. However, the evolution of HII regions around
stars begins in dense star-forming clouds and spans large dynamical ranges in
both space and time, posing severe challenges for numerical simulations in
terms of both spatial and temporal resolution that depends strongly on gas
density ($\propto n^{-1}$). In this work, we perform a series of idealized HII
region simulations using the moving-mesh radiation-hydrodynamic code Arepo-RT
to study the effects of numerical resolution. The simulated results match the
analytical solutions and the ionization feedback converges only if the
Str\"omgren sphere is resolved by at least $10$--$100$ resolution elements and
the size of each time integration step is smaller than $0.1$ times the
recombination timescale. Insufficient spatial resolution leads to reduced
ionization fraction but enhanced ionized gas mass and momentum feedback from
the HII regions, as well as degrading the multi-phase interstellar medium into
a diffuse, partially ionized, warm ($\sim8000$ K) gas. On the other hand,
insufficient temporal resolution strongly suppresses the effects of ionizing
feedback. This is because longer timesteps are not able to resolve the rapid
variation of the thermochemistry properties of the gas cells around massive
stars, especially when the photon injection and thermochemistry are performed
with different cadences. Finally, we provide novel numerical implementations to
overcome the above issues when strict resolution requirements are not
achievable in practice.
