Convergence of the Space-Time Expansion Discontinuous Galerkin Method for Scalar Conservation Laws

In this paper we analyse a class of fully discrete Space-Time Expansion Discontinuous-Galerkin methods for scalar conservation laws. This method has been introduced in [11, 17, 18] for a speci c expansion relying on the Cauchy-Kovaleskaya technique. We introduce a general concept of admissible expansions which in particular allows us to prove an error estimate for smooth solutions. The result applies for ansatz functions of arbitrary polynomial order k ∈ N provided the time step is su ciently small. It gives a convergence rate of order k + 12 in space and time. Finally we show that the Cauchy-Kovaleskaya technique leads to an admissible exansion. Furthermore we introduce two new expansions and prove that one of them, the characteristic expansion, is also admissible.