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We propose a spectral viscosity method to approximate the two-dimensional Euler equations with rough initial data and prove that the method converges to a weak solution for a large class of initial data, including when the initial vorticity is in the so-called Delort class, i.e., it is a sum of a signed measure and an integrable function. This provides the first convergence proof for a numerical method approximating the Euler equations with such rough initial data and closes the gap between the available existence theory and rigorous convergence results for numerical methods. We also present numerical experiments, including computations of vortex sheets and confined eddies, to illustrate the proposed method., Author(s): Samuel Lanthaler [sup.1] , Siddhartha Mishra [sup.1] Author Affiliations: (1) grid.5801.c, 0000 0001 2156 2780, Seminar for Applied Mathematics, Department of Mathematics, ETH Zurich, , Rämistrasse 101, 8092, Zurich, [...]

Entropy solutions have been widely accepted as the suitable solution framework for systems of conservation laws in several space dimensions. However, recent results in De Lellis and Székelyhidi Jr (Ann Math 170(3):1417-1436, 2009 (See CR17)) and Chiodaroli et al. (2013 (See CR18)) have demonstrated that entropy solutions may not be unique. In this paper, we present numerical evidence that state-of-the-art numerical schemes need not converge to an entropy solution of systems of conservation laws as the mesh is refined. Combining these two facts, we argue that entropy solutions may not be suitable as a solution framework for systems of conservation laws, particularly in several space dimensions. We advocate entropy measure-valued solutions, first proposed by DiPerna, as the appropriate solution paradigm for systems of conservation laws. To this end, we present a detailed numerical procedure which constructs stable approximations to entropy measure-valued solutions, and provide sufficient conditions that guarantee that these approximations converge to an entropy measure-valued solution as the mesh is refined, thus providing a viable numerical framework for systems of conservation laws in several space dimensions. A large number of numerical experiments that illustrate the proposed paradigm are presented and are utilized to examine several interesting properties of the computed entropy measure-valued solutions., Author(s): Ulrik S. Fjordholm [sup.1] , Roger Käppeli [sup.2] , Siddhartha Mishra [sup.2] , Eitan Tadmor [sup.3] Author Affiliations: (1) 0000 0001 1516 2393grid.5947.fDepartment of Mathematical Sciences, Norwegian University of [...]

We prove that the ENO reconstruction and ENO interpolation procedures are stable in the sense that the jump of the reconstructed ENO point values at each cell interface has the same sign as the jump of the underlying cell averages across that interface. Moreover, we prove that the size of these jumps after reconstruction relative to the jump of the underlying cell averages is bounded. Similar sign properties and the boundedness of the jumps hold for the ENO interpolation procedure. These estimates, which are shown to hold for ENO reconstruction and interpolation of arbitrary order of accuracy and on nonuniform meshes, indicate a remarkable rigidity of the piecewise polynomial ENO procedure. Keywords Newton interpolation * Adaptivity * ENO reconstruction * Sign property Mathematics Subject Classification 65D05 * 65M12, Introduction and Statement of Main Results The acronym ENO in the title of this paper stands for 'essentially non- oscillatory,' and it refers to a reconstruction procedure that generates a [...]