Atmospheric mixing ratios of the hydroxyl radical have relatively short lifespans (on the order of microseconds). When modeling air quality or the weather, the time step is usually much larger than the the half life of hydroxyl. How is this computation numerically stable? Is the problem of numerical stability simply ignored?
The hydroxyl radical chemistry is not modeled explicitly in air quality models, so numerical stability is not an issue. Instead, OH is held in pseudo-steady-state. For instance, see CMAQ's documentation which states:
In CMAQ's gas phase, OH is assumed to be in pseudo-steady state, and it is not a transported species. This is because hydroxyl radical reactions tend to be catalytic (e.g., consumption and production). In the aqueous phase chemistry, OH is absorbed by cloud water and in an open cloud model (i.e., the design approach currently employed), absorbed species (e.g., OH) would be replenished via gas-to-cloud partitioning. However, due to operator splitting, aqueous and gas-phase chemistry are not solved simultaneously. To account for this and other uncertainties in predicted OH aqueous phase concentrations (e.g., neglect of production reactions (H2O2 + hv -> 2 OH) not currently implemented in aqchem), a steady-state assumption for OH is adopted in the aqueous chemistry routine.
Typically, if you are trying to model something like the hydroxyl radical, you would use a box-model like MECCA, not a chemical transport model.
In fact, there are many chemical transport models which solve the radical chemistry without steady-state approximations. The equations are very stiff but can be integrated with suitable methods, usually implicit. Some approaches are discussed in this review. The Kinetic Preprocessor, mentioned in the comment, includes a set solvers for stiff systems and has become somewhat popular in atmospheric chemistry. Some models that use code generated by it (either by default or as an option) include GEOS-Chem and WRF-Chem. I used KPP to implement the CB4 chemical mechanism in this air quality model.
An approach like this is possible because of operator splitting, which means that the processes included in the model are solved as a sequence of uncoupled operations. For example, a model time step could consist of calling an explicit scheme for solving advection, then Crank-Nicolson for vertical diffusion, and then a Rosenbrock solver for chemical kinetics.