A Theory of Implicit Numerical Methods for Scalar Conservation Laws

We present a novel theory of implicit methods for scalar conservation laws which does to a large extent not rely on classical structures in numerical analysis, e.g. the notion of total variation stability is not employed. Thereby, for the first time rigorous theoretical results concerning the range of applicability of implicit schemes are given. A new proof of convergence is given relying on monotonicity of the considered methods and on a special discretization of the initial data. The theoretical results are confirmed by numerical tests