Some initial ideas will be presented for obtaining higher than second order accurate fluctuation splitting schemes for time-dependent problems (so the additional accuracy is in both space and time). Much of this work was done as part of an MSc project by Alistair Laird at the University of Reading.

 The resulting (currently linear) fluctuation distribution scheme achieve third order spatial accuracy by extending the range of grid nodes to which the fluctuation within each cell is distributed. The method is extremely compact and efficient, with cpu-time and memory overheads comparable to those of currently used second order schemes.

 The improved accuracy in time is obtained either by applying a third order Runge-Kutta discretisation to the time derivative, or by implementing a third order version of the Lax-Wendroff `trick' and, using the same stencil as the basic scheme, inserting approximations to the resulting second and third spatial derivative terms of the appropriate order of accuracy.

 The stability, accuracy and (non-)positivity of these schemes, as applied to the two-dimensional scalar advection equation on triangular meshes, will be discussed, along with their current limitations and potential for development.