Numerical Analysis of Flux Reconstruction
Creators: name: Trojak, William
Publisher: name: Apollo - University of Cambridge Repository
Year: 2020
Subjects: High-order methods
computational fluid dynamics
flux reconstruction
numerical analysis

description: High-order methods have become of increasing interest in recent years in computational physics. This is in part due to their perceived ability to, in some cases, reduce the computational overhead of complex problems through both an efficient use of computational resources and a reduction in the required degrees of freedom. One such high-order method in particular – Flux Reconstruction – is the focus of this thesis. This body of work relies and expands on the theoretical methods that are used to understand the behaviour of numerical methods – particularly related to their real-world application to industrial problems. The thesis begins by challenging some of the existing dogma surrounding computational fluid dynamics by evaluating the performance of high-order flux reconstruction. First, the use of the primitive variables as an intermediary step in the construction of flux terms is investigated. It is found that reducing the order of the flux function by using the conserved rather than primitive variables has a substantial impact on the resolution of the method. Critically, this is supported by a theoretical analysis, which shows that this mechanism of error generation becomes increasing important to consider as the order of accuracy increases. Next, the analysis of Flux Reconstruction was extended by analytically and numerically exploring the impact of higher dimensionality and grid deformation. It is found that both expanding and contracting grids – essential components of real-world domain decomposition – can cause dispersion overshoot in two dimensions, but that FR appears to suffer less that comparable Finite Difference approaches. Fully discrete analysis is then used to show that, depending on the correction function, small perturbations in incidence angle can cause large changes in group velocity. The same analysis is also used to theoretically demonstrate that Discontinuous Galerkin suffers less from dispersion error than Huynh’s FR scheme – a phenomenon that has previously been observed experimentally, but not explained theoretically. This thesis concludes with the presentation of a robust theoretical underpinning for determining stable correction functions for FR. Three new families of correction functions are presented, and their properties extensively explored. An important theoretical finding is introduced – that stable correction functions are not defined uniquely be a norm. As a result, a generalised approach is presented, which is able to recover all previously defined correction functions, but in some instances via a different norm to their original derivation. This new super-family of correction functions shows considerable promise in increasing temporal stability limits, reducing dispersion when fully discretised, and increasing the rate of convergence. Taken altogether, this thesis represents a considerable advance in the theoretical characterisation and understanding of a numerical method – one which, it has been shown, has enormous potential for forming the heart of future computational physics codes.