  We develop a nonlocal-response generalization to the Green-function
surface-integral method (GSIM), also known as the boundary-element method
(BEM). This numerically light method can accurately describe the linear
hydrodynamic nonlocal response of arbitrarily shaped plasmonic nanowires in
arbitrary dielectric backgrounds. All previous general-purpose methods for
nonlocal response are bulk methods. We also expand the possible geometries to
which the usual local-response GSIM can be applied, by showing how to
regularize singularities that occur in the surface integrals when the
nanoparticles touch a dielectric substrate. The same regularization works for
nonlocal response. Furthermore, an effective theory is developed to explain the
numerically observed nonlocal effects. The nonlocal frequency blueshift of a
cylindrical nanowire in an inhomogeneous background generally increases as the
nanowire radius and the longitudinal wavenumber become smaller, or when the
effective background permittivity or the mode inhomogeneity increase. The
inhomogeneity can be expressed in terms of an effective angular momentum of the
surface-plasmon mode. We compare local and nonlocal response of free-standing
nanowires, and of nanowires close to and on top of planar dielectric
substrates. Especially for the latter geometry, considerable differences in
extinction cross sections are found for local as compared to nonlocal response,
similar to what is found for plasmonic dimer structures.
