Runge-Kutta residual distribution schemes

The residual distribution framework and its ability to carry out genuinely multidimensional upwinding has attracted a lot of research interest in the past three
decades. Although not as robust as other widely used approximate methods for solving hyperbolic partial differential equations, when residual distribution schemes
do provide a plausible solution it is usually more accurate than in the case of other approaches. Extending these methods to time-dependent problems remains one of the main challenges in the field. In particular, constructing such a solution so that the resulting discretisation exhibits all the desired properties available in the steady state setting.
It is generally agreed that there is not yet an ideal generalisation of second order accurate and positive compact residual distribution schemes designed within the steady residual distribution framework to time-dependent problems. Various approaches exist, none of which is considered optimal nor completely satisfactory. In this thesis two possible extensions are constructed, analysed and verified numerically: continuous-in-space and discontinuous-in-space Runge-Kutta Residual Distribution
methods. In both cases a Runge-Kutta-type time-stepping method is used to integrate the underlying PDEs in time. These are then combined with, respectively, a continuous- and discontinuous-in-space residual distribution type spatial approximation.
In this work a number of second order accurate linear continuous-in-space Runge- Kutta residual distribution methods are constructed, tested experimentally and compared
with existing approaches. Additionally, one non-linear second order accurate scheme is presented and verified. This scheme is shown to perform better in the close vicinity of discontinuities (in terms of producing spurious oscillations) when compared to linear second order schemes. The experiments are carried out on a set of structured and unstructured triangular meshes for both scalar linear and nonlinear equations, and for the Euler equations of fluid dynamics as an example of systems of non-linear equations.
In the case of the discontinuous-in-space Runge-Kutta residual distribution framework, the thorough analysis presented here highlights a number of shortcomings of this approach and shows that it is not as attractive as initially anticipated. Nevertheless, a rigorous overview of this approach is given. Extensive numerical results on
both structured and unstructured triangular meshes confirm the analytical results. Only results for scalar (both linear and non-linear) equations are presented.