  Numerous tabletop experiments have been dedicated to exploring the
manifestations of screened scalar field dark energy, such as symmetron or
chameleon fields. Precise theoretical predictions require simulating field
configurations within the respective experiments. This paper focuses onto the
less-explored environment-dependent dilaton field, which emerges in the strong
coupling limit of string theory. Due to its exponential self-coupling, this
field can exhibit significantly steeper slopes compared to symmetron and
chameleon fields, and the equations of motion can be challenging to solve with
standard machine precision. We present the first exact solution for the
geometry of a vacuum region between two infinitely extended parallel plates.
This solution serves as a benchmark for testing the accuracy of numerical
solvers. By reparametrizing the model and transforming the equations of motion,
we show how to make the model computable across the entire experimentally
accessible parameter space. To simulate the dilaton field in one- and
two-mirror geometries, as well as spherical configurations, we introduce a
non-uniform finite difference method. Additionally, we provide an algorithm for
solving the stationary Schr\"odinger equation for a fermion in one dimension in
the presence of a dilaton field. The algorithms developed here are not limited
to the dilaton field, but can be applied to similar scalar-tensor theories as
well. We demonstrate such applications at hand of the chameleon and symmetron
field. Our computational tools have practical applications in a variety of
experimental contexts, including gravity resonance spectroscopy (qBounce),
Lunar Laser Ranging (LLR), and the upcoming Casimir and Non-Newtonian Force
Experiment (CANNEX). A Mathematica implementation of all algorithms is
provided.
