  This paper presents a simple and highly accurate method for capturing sharp
interfaces moving in divergence-free velocity fields using the high-order Flux
Reconstruction approach on unstructured grids. A well-known limitation of
high-order methods is their susceptibility to the Gibbs phenomenon; the
appearance of spurious oscillations in the vicinity of discontinuities and
steep gradients makes it difficult to accurately resolve shocks or sharp
interfaces. In order to address this issue in the context of sharp interface
capturing, a novel, preconditioned and localized phase field method is
developed in this work. The numerical accuracy of interface normal vectors is
improved by utilizing a preconditioning procedure based on the level set method
with localized artificial viscosity stabilization. The developed method was
implemented in the framework of the multi-platform Flux Reconstruction
open-source code PyFR. Numerical tests in 2D and 3D conducted on different mesh
types showed that the preconditioning procedure significantly improves
accuracy. The results demonstrate the conservativeness of the proposed method
and its ability to capture highly distorted interfaces with superior accuracy
when compared to conventional and high-order VOF and level set methods. The
high accuracy and locality of the proposed method offer a promising route to
carrying out massively-parallel, high accuracy simulations of multi-phase,
incompressible phenomena.
