Computational fluid dynamics has progressed to the point where it is now possible to simulate
flows with large eddy turbulence, free surfaces and other complex features. However, the success
of these models often depends on the accuracy of the advection scheme supporting them.
Two such schemes are the constrained interpolation profile method (CIP) and the interpolated
differential operator method (IDO). They share the same space discretisation but differ in their
respectively semi-Lagrangian and Eulerian formulations. They both belong to a family of high-order,
compact methods referred to as the multi-moment methods.
In the absence of sufficient information in the literature, this thesis begins by taxonomising
various multi-moment space discretisations and appraising their linear advective properties. In
one dimension it is found that the CIP/IDO with order (2N -1) has an identical spectrum and
memory cost to the Nth order discontinuous Galerkin method. Tests confirm that convergence
rates are consistent with nominal orders of accuracy, suggesting that CIP/IDO is a better choice
for smooth propagation problems. In two dimensions, six Cartesian multi-moment schemes of
third order are compared using both spectral analysis and time-domain testing. Three of these
schemes economise on the number of moments that need to be stored, with one CIP/IDO
variant showing improved isotropy, another failing to maintain its nominal order of accuracy,
and one of the conservative variants having eigenvalues with positive real parts: it is stable only
in a semi-Lagrangian formulation. These findings should help researchers who are interested
in using multi-moment schemes in their solvers but are unsure as to which are suitable.
The thesis then addresses the question as to whether a multi-moment method could be implemented
on a Cartesian cut cell grid. Such grids are attractive for supporting arbitrary, possibly
moving boundaries with minimal grid regeneration. A pair of novel conservative fourth order
schemes is proposed. The first scheme, occupying the Cartesian interior, has unprecedented
low memory cost and is proven to be conditionally stable. The second, occupying the cut cells,
involves a profile reconstruction that is guaranteed to be well-behaved for any shape of cell.
However, analysis of the second scheme in a simple grid arrangement reveals positive real
parts, so it is not stable in an Eulerian formulation. Stability in a hybrid formulation remains
open to question
