Modelling and Analysis for Curvature Driven Partial Differential Equations

This thesis consists of two parts. The first part studies a family of monotone finite volume schemes for a scalar hyperbolic conservation law on a closed Riemannian manifold M of arbitrary dimension. For initial data with bounded total variation on M we will prove that these schemes converge with a h1/4 convergence rate in L1 towards the entropy solution. When M is 1-dimensional the schemes are total variation diminishing and we will show that this improves the convergence rate to h1/2.

The second part deals with a sharp interface limit of the isothermal Navier-Stokes- Korteweg equations. The sharp interface limit is performed by matched asymptotic ex- pansions of the fields in powers of the interface width. We consider two sets of expansions, one in the interfacial region (inner expansions) and one in the bulk (outer expansions) and match them order by order. For a specific scaling we prove solvability conditions for the first orders of the inner equations, which are obtained by a change of coordinates in the vicinity of the interface. These solvability criteria result in jump conditions for the bulk fields. These jump conditions fit into the general framework for jump conditions for sharp interface models.