Discontinuous fluctuation distribution

This paper describes a new numerical scheme for the approximation of steady state solutions to systems of hyperbolic conservation laws. It generalises the fluctuation distribution framework by allowing the underlying representation of the solution to be discontinuous. This leads to edge-based fluctuations in addition to the standard cell-based fluctuations, which are then distributed to the cell vertices in an upwind manner which retains the properties of the continuous scheme (positivity, linearity preservation, conservation, compactness and continuity). Numerical results are presented on unstructured triangular meshes in two space dimensions for linear and nonlinear scalar equations as well as the Euler equations of gasdynamics. The accuracy of the approximation in smooth regions of the flow is shown to be very similar to the corresponding continuous scheme, but the discontinuous approach improves the sharpness with which discontinuities in the flow can be captured and provides additional flexibility which will allow adaptive techniques to be applied simply to improve efficiency.