High-order methods for computational fluid dynamics: A brief review of compact differential formulations on unstructured grids

•Discontinuous Galerkin (DG) and Flux Reconstruction (FR) are explained for an integration problem.•Flux Reconstruction/Correction Procedure via Reconstruction (FR/CPR) schemes are derived for conservation laws.•2D and 3D extensions of FR/CPR methods are reviewed.•Stability proofs for these schemes are sketched.•Recent research and some pacing items for this approach are described.

Popular high-order schemes with compact stencils for Computational Fluid Dynamics (CFD) include Discontinuous Galerkin (DG), Spectral Difference (SD), and Spectral Volume (SV) methods. The recently proposed Flux Reconstruction (FR) approach or Correction Procedure using Reconstruction (CPR) is based on a differential formulation and provides a unifying framework for these high-order schemes. Here we present a brief review of recent progress in FR/CPR research as well as some pacing items and future challenges.