On the development and implementation of high-order flux reconstruction schemes for computational fluid dynamics

High-order numerical methods for unstructured grids combine the superior accuracy of high-order spectral or finite difference methods with the geometric 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 graphics 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 within the vicinity of real-world engineering geometries. In this thesis a formulation of the FR approach that is suitable for solving non-linear advection-diffusion type problems on mixed curvilinear grids is developed. Issues around aliasing are explored in detail and techniques for mitigation outlined. A methodology for identifying symmetric quadrature rules inside of a variety of domains is also presented and used to find several rules that appear to be an improvement over those in literature. This methodology is also used to obtain improved sets of solution points inside of triangular elements. PyFR, an open-source Python based framework for solving the compressible Navier–Stokes equations using the FR approach, is also developed. It is designed to target a range of hardware platforms via use of an in-built domain specific language based on the Mako templating engine. PyFR is able to operate on mixed grids in both two and three dimensions and can target NVIDIA GPUs, AMD GPUs, and Intel CPUs. Results are presented for various benchmark flow problems, single-node performance is discussed, heterogeneous multi-node capabilities are analysed, and scalability is demonstrated on up to 2000 NVIDIA K20X GPUs for a sustained performance of 1.3 PFLOP/s.Open Acces