Abstract/Summary

Numerical weather prediction models are simulating more passive and active tracers than anytime in history. In the next generation of dynamical cores unconditionally monotone, stable, locally mass preserving algorithms are sought. Due to the importance of local mass conservation, the historically dominant semi-Lagrangian method may see competitors from the Eulerian frame. This thesis is concerned with the development of slope and flux limiters for dynamical core advection algorithms. The first investigation in this thesis regards the use and development of one-dimensional limiters applied to the multidimensional advection equation on a uniform grid. Two new limiter regions are derived which are sufficient for the numerical solution of the incompressible advection to retain a local maximum principle. The second investigation in this thesis regards the use and development of truly unstructured multidimensional slope limiters suitable for a wider class of schemes and meshes. A general theory is presented, with two limiters introduced capable of preserving different local maximum principles. We then illustrate two practical examples of how this theory can be applied. The first example introduces the limiters and how slight improvements on state-of-the-art limiters can be achieved for second order finite volume methods. The second example illustrates the true use of the theory by introducing a new fourth order finite volume scheme, and how the limiting procedures can be used for discrete maximum principles. The third contribution of this thesis consists of a numerical study in implicit linearised slope limiters and the use of implicit flux corrected transport. This approach concerns how one can achieve stability and monotonicity at large Courant numbers, whilst still retaining monotonicity and accuracy at low Courant numbers. A simple one stage flux corrected transport scheme on two implicit methods emerges as a robust method achieving many of the desired properties for a dynamical core advection algorithm.