Discontinuous Fluctuation Distribution Schemes for Hyperbolic Conservation Laws

It is now commonplace to use numerical methods to simulate complex multidimensional fluid flow problems, and the available computing power has increased to a point where, for example, it is routine to compute flows around complete aircraft configurations. These calculations can capture very fine levels of detail in the definition of the aircraft geometry, but the nonlinearities inherent in the mathematical model of the flow, and the lack of sophistication of many popular approximation techniques mean that including these finer levels of detail doesn't necessarily lead to greater accuracy in the simulation. The most widely-used numerical models are typically based on one-dimensional physical phenomena (representing the solution by a series of local Riemann problems aligned with the edges of the computational mesh) and fail to build genuinely multidimensional features into their design. It is easily shown that such models can misinterpret even the simplest of flows if they are not aligned with the mesh, so although the methods are extremely robust and often provide plausible representations of the large scale flow features, the fine detail cannot be relied upon to be accurate. Fluctuation distribution schemes have been specifically designed to address these issues by building genuinely multidimensional physical processes into the discrete form of the conservation laws. This improves the approximation of the fine detail (at the expense of some of the robustness) and consequently provides a more accurate representation of the global flow. Despite considerable success, the original framework was based on a continuous representation of the flow, and this constraint is a little restrictive. Allowing discontinuities provides more flexibility and has the potential to improve both the accuracy in the simulation of genuinely discontinuous flows and the efficiency of the method, especially since it simplifies the application of adaptive techniques. For time-dependent problems, there is an additional benefit in that the discontinuous representation can be used to avoid the expensive step of inverting a ``mass matrix'' at every time-step. A framework for discontinuous fluctuation distribution has recently been proposed, which combines multidimensional physical modelling, inherited from the continuous upwind fluctuation distribution schemes, with the flexibility of the discontinuous representation. This project will attempt to exploit the potential of this approach and apply it to a wide range of fluid flow problems.