From the release notes:
High-order numerical methods for flow simulations capture complex phenomena like vortices and separation regions using fewer degrees of freedom than their low-order counterparts. The High Fidelity (HiFi) provided by the schemes, combined with turbulence models for small scales and wall interactions, gives rise to a powerful Large Eddy Simulation (LES) software package. HiFiLES is an open-source, high-order, compressible flow solver for unstructured grids built from the ground up to take full advantage of parallel computing architectures. It is specially well-suited for Graphical Processing Unit (GPU) architectures. HiFiLES is written in C++. The code uses the MPI protocol to run on multiple processors, and CUDA to harness GPU performance.
The main reference for the code right now is this V&V paper. The code uses an Energy Stable Flux Reconstruction (ESFR) scheme. Here are a couple papers on that approach.[2, 3].
 López-Morales, M. R., Bull, J., Crabill, J., Economon, T. D., Manosalvas, D., Romero, J., Sheshadri, A., Watkins II, J. E., Williams, D., Palacios, F., et al., “Verification and Validation of HiFiLES: a High-Order LES unstructured solver on multi-GPU platforms,” .
 Vincent, P. E., Castonguay, P., and Jameson, A., “A new class of high-order energy stable flux reconstruction schemes,” Journal of Scientific Computing, Vol. 47, No. 1, 2011, pp. 50–72.
 Castonguay, P., Vincent, P. E., and Jameson, A., “A new class of high-order energy stable flux reconstruction schemes for triangular elements,” Journal of Scientific Computing, Vol. 51, No. 1, 2012, pp. 224–256.