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

摘要

We study a recent timestep-adaptation technique for hyperbolic conservation laws. The key tool is a space-time splitting of adjoint error representations for target functionals due to Sueli (An Introduction to Recent Developments in Theory and Numerics for Conservation Laws. Lecture Notes in Computational Science and Engineering. Springer: Berlin, 1998; 123-194) and Hartmann (A posteriori Fehlerschaetzung und adaptive Schrittweiten- und Ortsgittersteuerung bei Galerkin-Verfahren fur die Warmeleitungsgleichung. Diplomarbeit, Institut fuer Angewandte Mathematik, Universitat Heidelberg, 1998). It provides an efficient choice of timesteps for implicit computations of weakly instationary flows. The timestep will be very large in regions of stationary flow and become small when a perturbation enters the flow field. Besides using adjoint techniques that are already well established, we also add a new ingredient that simplifies the computation of the dual problem. Owing to Galerkin orthogonality, the dual solution φ does not enter the error representation as such. Instead, the relevant term is the difference of the dual solution and its projection to the finite element space, φ - φ_h. We can show that it is therefore sufficient to compute the spatial gradient of the dual solution, w = ▽φ. This gradient satisfies a conservation law instead of a transport equation, and it can therefore be computed with the same algorithm as the forward problem, and in the same finite element space. We demonstrate the capabilities of the approach for a weakly instationary test problem for scalar conservation laws.