The major issue addressed in this thesis is the development of a Mach-uniform method that uses a staggered scheme on planar unstructured grids with triangular cells, that has a superlinear rate of convergence. In our staggered grid, the normal velocity components are associated with cell faces, whereas scalar variables are assigned to cell centers. In order to obtain a superlinearly convergent scheme, it is necessary to interpolate vector fields from staggered components with sufficient accuracy in a robust way. Several methods to achieve this objective are investigated. Like all higher order methods, the superlinearly convergent scheme obtained suffers from unphysical oscillations near discontinuities such as shocks. This difficulty is overcome by combining two approaches. We use an upwind-biased superlinearly convergent scheme, so that oscillations are damped in smooth parts of the flow. Furthermore, we follow the usual flux limiting approach to combat oscillations near discontinuities, extended to unstructured staggered schemes. An oscillation detecting limiter function is used to switch between the superlinearly convergent scheme and a first order scheme that generates (almost) no oscillations. For this we adapt the limiter of Barth and Jespersen.@en