Abstract: 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 specific 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 sufficiently small. It gives a convergence rate of order k+½ in space and time. Finally we show that the original Cauchy–Kovaleskaya technique leads to an admissible expansion. Furthermore we introduce two new expansions and prove that one of them, the characteristic expansion, is also admissible.

[11] Gassner, G.; Lörcher, F.; Munz, C.-D.: A discontinuous Galerkin scheme based on a space-time expansion. II. Viscous flow equations in multi dimensions. J. Sci. Comput. 34 (2008), Nr. 3, S. 260–286

[17] Lörcher, F.; Gassner, G.; Munz, C.-D.: A discontinuous Galerkin scheme based on a space-time expansion. I. Inviscid compressible flow in one space dimension. J. Sci. Comput. 32 (2007), Nr. 2, S. 175–199

[18] Lörcher, F.; Gassner, G.; Munz, C.-D.: An explicit discontinuous Galerkin scheme with local time-stepping for general unsteady diffusion equations. J. Comput. Phys. 227 (2008), Nr. 11, S. 5649–5670