On the numerical solution of multi-dimensional non-linear systems of conservation laws

The paper describes a general framework for the construction and analysis of numerical methods for systems of conservation laws with focus on multi-dimensional equations. The basic ideas are related to the Method of Transport originally derived for the Euler equations. The continuous extension of this approach to different systems and other methods results in a general description of genuinely multi-dimensional methods in terms of wave propagation. Many of the existing multi-dimensional schemes can be put into the resulting framework, which allows a comparison on a more analytic level. We first derive numerical methods in a very general setting. Next, we take the Euler equations as an example to demonstrate the use of this approach by means of the Method of Transport and the Steger-Warming Awe-vector splitting. Finally we include fluctuation splitting schemes into this setting and show possible extensions.