The Space-Time Expansion DG Method
In this paper the recently developed space-time expansion discontinuous Galerkin (STE-DG) approach for the two dimensional unsteady compressible Navier-Stokes equations is presented. The basis of the scheme is a weak formulation of the Navier-Stokes equations, where special care of the second order terms is taken. The spatial polynomial of the DG approach is expanded in time using the so called Cauchy-Kovalevskaya (CK) procedure. With a polynomial of order N in space the CK procedure generates an approximation of order N in time as well yielding a scheme of order N+1 in space and time. The locality and the space-time nature of the presented method give the interesting feature that the time steps may be different in each grid cell. Hence, we drop the common global time steps and propose for a time-accurate solution that any grid cell runs with its own time step determined by the local stability restriction. In spite of the local time steps the scheme is conservative, fully explicit, and as in the DG approach the polynomial order could be chosen arbitrarily, the scheme is theoretically of arbitrary order of accuracy in space and time for transient calculations.
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