Theory

High-order numerical methods for unstructured grids combine the
superior accuracy of high-order spectral or finite difference methods
with the geometrical flexibility of low-order finite volume or finite
element schemes. The Flux Reconstruction (FR) approach unifies various
high-order schemes for unstructured grids within a single framework.
Additionally, the FR approach exhibits a significant degree of element
locality, and is thus able to run efficiently on modern streaming
architectures, such as Graphical Processing Units (GPUs). The
aforementioned properties of FR mean it offers a promising route to
performing affordable, and hence industrially relevant, scale-resolving
simulations of hitherto intractable unsteady flows (involving
separation, acoustics etc.) within the vicinity of real-world
engineering geometries. An detailed overview of the FR approach is
given in:

The linear stability of an FR schemes depends on the form of the
correction function. Linear stability issues are discussed in:

- A New Class of High-Order Energy Stable Flux Reconstruction Schemes. P. E. Vincent, P. Castonguay, A. Jameson. Journal of Scientific Computing, Volume 47, Number 1, Pages 50-72, 2011
- Insights from von Neumann Analysis of High-Order Flux Reconstruction Schemes. P. E. Vincent, P. Castonguay, A. Jameson. Journal of Computational Physics, Volume 230, Issue 22, Pages 8134-8154, 2011
- A New Class of High-Order Energy Stable Flux Reconstruction Schemes for Triangular Elements. P. Castonguay, P. E. Vincent, A. Jameson. Journal of Scientific Computing, Volume 51, Number 1, Pages 224-256, 2012
- Energy Stable Flux Reconstruction Schemes for Advection-Diffusion Problems. P. Castonguay, D. M. Williams, P. E. Vincent, A. Jameson. Computer Methods in Applied Mechanics and Engineering, Volume 267, Pages 400-417, 2013
- Energy Stable Flux Reconstruction Schemes for Advection-Diffusion Problems on Triangles. D. M. Williams, P. Castonguay, P. E. Vincent, A. Jameson. Journal of Computational Physics, Volume 250, Pages 53-76, 2013
- Energy Stable Flux Reconstruction Schemes for Advection-Diffusion Problems on Tetrahedra. D. M. Williams, A. Jameson. Journal of Scientific Computing, Volume 59, Pages 721-759, 2014
- An Extended Range of Stable-Symmetric-Conservative Flux Reconstruction Correction Functions. P. E. Vincent, A. M. Farrington, F. D. Witherden, A. Jameson. Computer Methods in Applied Mechanics and Engineering, Volume 296, Pages 248-272, 2015

The non-linear stability of an FR schemes depends on the location of the
solution points. Non-linear stability issues are discussed in:

- On the Non-Linear Stability of Flux Reconstruction Schemes. A. Jameson, P. E. Vincent, P. Castonguay. Journal of Scientific Computing, Volume 50, Number 2, Pages 434-445, 2012
- An Analysis of Solution Point Coordinates for Flux Reconstruction Schemes on Triangular Elements. F. D. Witherden, P. E. Vincent. Journal of Scientific Computing, Volume 61, Pages 398-423, 2014