  Second-order self-force computations, which will be essential in modeling
extreme-mass-ratio inspirals, involve two major new difficulties that were not
present at first order. One is the problem of large scales, discussed in [Phys.
Rev. D 92, 104047 (2015)]. Here we discuss the second difficulty, which occurs
instead on small scales: if we expand the field equations in spherical
harmonics, then because the first-order field contains a singularity, we
require an arbitrarily large number of first-order modes to accurately compute
even a single second-order mode. This is a generic feature of nonlinear field
equations containing singularities, allowing us to study it in the simple
context of a scalar toy model in flat space. Using that model, we illustrate
the problem and demonstrate a robust strategy for overcoming it.
