Numerical Analysis of Flux Reconstruction

William Trojak, Paul Tucker, Stewart Cant, Apollo-University Of Cambridge Repository, Apollo-University Of Cambridge Repository
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
more » ... ds 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, bu [...]
doi:10.17863/cam.51695 fatcat:wrs7x5dfwzeubjtsxa5o45bmtm