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Adaptive Timestep Control for Nonstationary Solutions of the Euler Equations

In this paper we continue our work on adaptive timestep control for weakly nonstationary problems [C. Steiner and S. Noelle, Internat. J. Numer. Methods Biomedical Engrg., 26 (2010), pp. 790-806]. The core of the method is a space-time splitting of adjoint error representations for target functionals due to Suli [A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems, in An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, Lect. Notes Comput. Sci. Eng. 5, Springer, Berlin, 1999, pp. 123-194] and Hartmann [Master's thesis, Universitat Heidelberg, Heidelberg, Germany, 1998]. The main new ingredients are (i) the extension from scalar, one-dimensional (1D), conservation laws to the two-dimensional (2D) Euler equations of gas dynamics, (ii) the derivation of boundary conditions for a new formulation of the adjoint problem, and (iii) the coupling of the adaptive time-stepping with spatial adaptation. For the spatial adaptation, we use a multiresolution-based strategy developed by Muller [Adaptive Multiscale Schemes for Conservation Laws, Lect. Notes Comput. Sci. Eng. 27, Springer, Berlin, 2003], and we combine this with an implicit time discretization. The combined space-time adaptive method provides an efficient choice of timesteps for implicit computations of weakly nonstationary flows. The timestep will be very large in time intervals of stationary flow and becomes small when a perturbation enters the flow field. The efficiency of the solver is investigated by means of an unsteady inviscid 2D flow over a bump.