Remapping (conservative interpolation), within arbitrary Lagrangian-Eulerian (ALE) schemes, requires the values of the scalar to be interpolated
from one computational mesh to another which has differing geometry.
Advection methods are typically utilised for the remapping stage, with
fluxes being created by overlapping volumes between adjacent elements.
In the thesis, a second-order, conservative, sign-preserving remapping
scheme is developed utilising concepts of the Multidimensional Positive
Definite Advection Transport Algorithm (MPDATA). The basic non-oscillatory and non-oscillatory infinite gauge options are derived for remapping
in volume co-ordinates. For the first time, an MPDATA based remapping has been successfully implemented into full ALE schemes. Inherent properties of MPDATA are exploited to reduce wall heating errors via the second-order filtering option. The resulting increase in accuracy and symmetry of numerical solutions is demonstrated. For material interfaces, an adaptive mixed cell approach is proposed which takes advantage of the efficient computational stencil of MPDATA. The proposed approach utilises
all available data in the calculation of pseudo velocities in MDPATA in order to retain second-order accuracy and multi-dimensionality at material interfaces. The effectiveness of the adaptive mixed cell approach is highlighted via examples featuring artificial material interfaces.
Theoretical developments are supported by numerical testing. All test cases compare the accuracy of the MPDATA based schemes to a van Leer based scheme generalised to multiple dimensions via isotropic or Strang
split remapping. The results demonstrate the advantages of the fully multi-dimensional MPDATA remapping.
A Doctoral Thesis. Submitted in partial fulfillment of the requirements for the award of Doctor of Philosophy of Loughborough University.