Undercompressive shock waves arise in many physical processes which involve multiple phases. We propose a Finite Volume method in two space dimensions to approximate weak solutions of systems of hyperbolic or hyperbolic-elliptic conservation laws that contain undercompressive shock waves. The method can be seen as a generalization of the spatially one-dimensional and scalar approach in [C. Chalons, P. Engel and C. Rohde, SIAM J. Numer. Anal. 52 (2014) 554-579]. It relies on a moving mesh ansatz such that the undercompressive wave is represented as a sharp interface without any artificial smearing. It is proven that the method is locally conservative and recovers planar traveling wave solutions exactly. To demonstrate the efficiency and reliability of the new scheme we test it on scalar model problems and as an application on compressible liquid-vapour flow in two space dimensions.
The dissipation of potential energy of multiply charged Ar ions incident on Cu has been studied by complementary electron spectroscopy and calorimetry at charge states between 2 and 10 and kinetic energies between 100 eV and 1 keV. The emitted and deposited fractions of potential energy increase at increasing charge state, showing a significant jump for charge states q > 8 due to the presence of L-shell vacancies in the ion. Both fractions balance the total potential energy, thus rendering former hypotheses of a significant deficit of potential energy obsolete. The experimental data are reproduced by computer simulations based on the extended dynamic classical-over-the-barrier model.
We consider the classical line simplification problem subject to a given error bound but with additional topology constraints as they arise for example in the map rendering domain. While theoretically inapproximability has been proven for these problem variants, we show that in practice one can solve medium sized instances optimally using an integer linear programming approach and larger instances using an heuristic approach which for medium-sized real-world instances yields close-to-optimal results. Our approaches are evaluated on data sets which are synthetically generated, stem from the OpenStreetMap project [1], and the recent GISCup competition [3].
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