Recent Developments in High-Order-Accurate Fluctuation Splitting Schemes

This paper provides some recent achievements in developing high-order-accurate fluctuation splitting (FS) schemes, by providing a review of some recent developments as well as a novel analysis on Lagrangian elements. The sufficient conditions for an FS scheme to achieve (r + 1)th-order accuracy in both space and time are derived at first. Then, the two suitable approaches for designing third-orderaccurate FS schemes are considered. The gradient based approach is analyzed at first. For a scalar conservation law, a well posed non-compact stable third-orderaccurate FS scheme is designed and validated versus a well known test case. The scheme is then generalized to the case of the Euler equations where, again, it is proven to achieve third-order accuracy in both space and time. The approach based on Lagrangian elements is then analyzed in details, only for the case of scalar advection. The third-order-accurate FS scheme, obtained by exact analytical integration, is shown to be consistent and conservative, but leads to an indeterminate linear system. A useful scheme can be obtained by applying the FS scheme to distribute the residual of each sub cell of the Lagrangian element among its three nodes