  This thesis focuses on the application of numerical relativity methods to the
solutions of problems in strong gravity. Our goal is the study of mergers of
compact objects in the strong field regime where non-linear dynamics manifest
and deviations from General Relativity could be found. We develop a new
formulation of the Einstein equations in d+1 spacetime dimensions in the moving
punctures approach, which leads to a well-posed set of equations for the
Einstein-Gauss-Bonnet gravity (EGB), as well as for the most general
parity-invariant scalar-tensor theory of gravity up to four derivatives
(4$\partial$ST). Using this formulation, we have implemented the equations of
the 4$\partial$ST theory in GRFolres, an open-source extension of our numerical
relativity code GRChombo. This has enabled us to evolve equal and unequal-mass
Binary Black Hole mergers in this effective field theory of gravity, as well as
to study the loss of hyperbolicity and its relation to the weak coupling
condition, among other topics of interest left for further study.
