Fast full $N$-body simulations of generic modified gravity: derivative
  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 two broad classes of MG models with derivative
coupling terms -- the Vainshtein- and Kmouflage-type models -- which
respectively features the Vainshtein and Kmouflage screening mechanism. 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 Kmouflage, we have proposed a new
algorithm for the relaxation solver, and run the first simulations of the model
to understand its cosmological behaviour. In a companion paper, we describe
versions of this code developed for conformally-coupled MG models, including
several variants of $f(R)$ gravity, the symmetron model and coupled
quintessence. Altogether, MG-GLAM has so far implemented the prototypes for
most MG models of interest, and is broad and versatile. The code is highly
optimised, with a tremendous (over two orders of magnitude) speedup when
comparing its running time 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
accurate emulators for ongoing and future galaxy surveys.
