$\texttt{fRevolution}$ $-$ Relativistic Cosmological Simulations in
  $f(R)$ Gravity I: Methodology
  We present the new relativistic cosmological particle-mesh code
$\texttt{fRevolution}$, based on $\texttt{gevolution}$, aimed at simulating
non-linear structure formation in $f(R)$ gravity. We introduce the general
framework and approximation scheme, and the set of equations used to solve for
the full set of gravitational perturbations. We show results for a point mass
field and for cosmological simulations in the Hu-Sawicki model, and compare
them to those of existing Newtonian codes. A more detailed analysis and
discussion of our solutions will be carried out in a following paper.
