A posteriori analysis of fully discrete method of lines DG schemes for systems of conservation laws

We present reliable a posteriori estimators for some fully discrete schemes applied to nonlinear systems of hyperbolic conservation laws in one space dimension with strictly convex entropy. The schemes are based on a method of lines approach combining discontinuous Galerkin spatial discretization with single- or multi-step methods in time. The construction of the estimators requires a reconstruction in time for which we present a very general framework first for odes and then apply the approach to conservation laws. The reconstruction does not depend on the actual method used for evolving the solution in time. Most importantly it covers in addition to implicit methods also the wide range of explicit methods typically used to solve conservation laws. For the spatial discretization, we allow for standard choices of numerical fluxes. We use reconstructions of the discrete solution together with the relative entropy stability framework, which leads to error control in the case of smooth solutions. We study under which conditions on the numerical flux the estimate is of optimal order pre-shock. While the estimator we derive is computable and valid post-shock for fixed meshsize, it will blow up as the meshsize tends to zero. This is due to a breakdown of the relative entropy framework when discontinuities develop. We conclude with some numerical benchmarking to test the robustness of the derived estimator

On the convergence of a shock capturing discontinuous Galerkin method for nonlinear hyperbolic systems of conservation laws

In this paper, we present a shock capturing discontinuous Galerkin (SC-DG) method for nonlinear systems of conservation laws in several space dimensions and analyze its stability and convergence. The scheme is realized as a space-time formulation in terms of entropy variables using an entropy stable numerical flux. While being similar to the method proposed in [14], our approach is new in that we do not use streamline diffusion (SD) stabilization. It is proved that an artificial-viscosity-based nonlinear shock capturing mechanism is sufficient to ensure both entropy stability and entropy consistency, and consequently we establish convergence to an entropy measure-valued (emv) solution. The result is valid for general systems and arbitrary order discontinuous Galerkin method.Comment: Comments: Affiliations added Comments: Numerical results added, shortened proo

Optimal Regularizing Effect for Scalar Conservation Laws

We investigate the regularity of bounded weak solutions of scalar conservation laws with uniformly convex flux in space dimension one, satisfying an entropy condition with entropy production term that is a signed Radon measure. The proof is based on the kinetic formulation of scalar conservation laws and on an interaction estimate in physical space.Comment: 24 pages, assumption (11) in Theorem 3.1 modified together with the example on p. 7, one remark added after the proof of Lemma 4.3, some typos correcte

Multidimensional Conservation Laws: Overview, Problems, and Perspective

Some of recent important developments are overviewed, several longstanding open problems are discussed, and a perspective is presented for the mathematical theory of multidimensional conservation laws. Some basic features and phenomena of multidimensional hyperbolic conservation laws are revealed, and some samples of multidimensional systems/models and related important problems are presented and analyzed with emphasis on the prototypes that have been solved or may be expected to be solved rigorously at least for some cases. In particular, multidimensional steady supersonic problems and transonic problems, shock reflection-diffraction problems, and related effective nonlinear approaches are analyzed. A theory of divergence-measure vector fields and related analytical frameworks for the analysis of entropy solutions are discussed.Comment: 43 pages, 3 figure