  There has been a lot of research interest in modified gravity theories which
utilise the Vainshtein mechanism to recover standard general relativity in
regions with high matter density, such as the Dvali-Gabadadze-Porrati and
Galileon models. The strong nonlinearity in the field equations of these
theories implies that accurate theoretical predictions could only be made using
high-resolution cosmological simulations. Previously, such simulations were
usually done on regular meshes, which limits both their performance and the
accuracy. In this paper, we report the development of a new algorithm and code,
based on ECOSMOG, that uses adaptive mesh refinements to improve the efficiency
and precision in simulating the models with Vainshtein mechanism. We have made
various code tests against the numerical reliability, and found consistency
with previous simulations. We also studied the velocity field in the
self-accelerating branch of the DGP model. The code, parallelised using MPI, is
suitable for large cosmological simulations of Galileon-type modified gravity
theories.
