A Framework for Discontinuous Fluctuation Distribution

The concept of fluctuation distribution was originally proposed nearly 25 years ago [6] as a potential alternative to flux-based finite volume schemes for approximating hyperbolic conservation laws. Major advances during the intervening period have provided genuinely multidimensional schemes which can achieve very high orders of accuracy without spurious oscillations, for both steady state and time-dependent scalar equations, along with generalisations to nonlinear systems of conservation laws. Details of both their foundations and some of the most important recent developments can be found in [7] (the reader is referred to the references therein for a full account). However, this has been matched by the progress of the finite volume approach, which has maintained its popularity for simulating flows of realistic complexity, largely due to its ability to provide plausible solutions in the most demanding of situations. Fluctuation distribution still lacks this robustness: its advantage is that when it does provide a sensible solution, it is typically more accurate, often by a significant margin, due its more realistic representation of multidimensional flow physics. One of the strongest challenges to the dominance of finite volume schemes has been the emergence of the discontinuous Galerkin approach (see, for example [2, 7]). This allies the discontinuous, edge-based form associated with finite volumes (in which the conserved quantity within a control volume varies according to the net flux through the surface of that volume) with the continuous, cell-based form associated with finite elements (in which the conserved quantity associated with a test function varies according to the local flux variations). The