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We study a recent timestep adaptation technique for hyperbolic conservationlaws. The key tool is a space-time splitting of adjoint error representationsfor target functionals due to S\"uli and Hartmann. It provides an efficientchoice of timesteps for implicit computations of weakly instationary flows. Thetimestep will be very large in regions of stationary flow, and become smallwhen a perturbation enters the flow field. Besides using adjoint techniqueswhich are already well-established, we also add a new ingredient whichsimplifies the computation of the dual problem. Due to Galerkin orthogonality,the dual solution {\phi} does not enter the error representation as such.Instead, the relevant term is the difference of the dual solution and itsprojection to the finite element space, {\phi}-{\phi}h . We can show that it istherefore sufficient to compute the spatial gradient of the dual solution, $w ={\nabla} {\phi}$. This gradient satisfies a conservation law instead of atransport equation, and it can therefore be computed with the same algorithm asthe forward problem, and in the same finite element space. We demonstrate thecapabilities of the approach for a weakly instationary test problem for scalarconservation laws.

On adaptive timestepping for weakly instationary solutions of hyperbolic conservation laws via adjoint error control