Fast full N-body simulations of generic modified gravity: conformal
  coupling models
  We present MG-GLAM, a code developed for the very fast production of full
$N$-body cosmological simulations in modified gravity (MG) models. We describe
the implementation, numerical tests and first results of a large suite of
cosmological simulations for three classes of MG models with conformal coupling
terms: the $f(R)$ gravity, symmetron and coupled quintessence models. Derived
from the parallel particle-mesh code GLAM, MG-GLAM incorporates an efficient
multigrid relaxation technique to solve the characteristic nonlinear partial
differential equations of these models. For $f(R)$ gravity, we have included
new variants to diversify the model behaviour, and we have tailored the
relaxation algorithms to these to maintain high computational efficiency. In a
companion paper, we describe versions of this code developed for derivative
coupling MG models, including the Vainshtein- and K-mouflage-type models.
MG-GLAM can model the prototypes for most MG models of interest, and is broad
and versatile. The code is highly optimised, with a tremendous speedup of a
factor of more than a hundred compared with earlier $N$-body codes, while still
giving accurate predictions of the matter power spectrum and dark matter halo
abundance. MG-GLAM is ideal for the generation of large numbers of MG
simulations that can be used in the construction of mock galaxy catalogues and
the production of accurate emulators for ongoing and future galaxy surveys.
