A Posteriori Analysis of Discontinuous Galerkin Schemes for Systems of Hyperbolic Conservation Laws

Abstract

Abstract. In this work we construct reliable a posteriori estimates for some discontinuous Galerkin schemes applied to nonlinear systems of hyperbolic conservation laws. We make use of appropriate reconstructions of the discrete solution together with the relative entropy stability framework.The methodology we use is quite general and allows for a posteriori control of discontinuous Galerkin schemes with standard flux choices which appear in the approximation of conservation laws.In addition to the analysis, we conduct some numerical benchmarking to test the robustness of the resultant estimator.1. Introduction. Hyperbolic conservation laws play an important role in many physical and engineering applications. One example is the description of non-viscous compressible flows by the Euler equations. Hyperbolic conservation laws in general only have smooth solutions up to some finite time even for smooth initial data. This makes their analysis and the construction of reliable numerical schemes challenging. The development of discontinuities poses significant challenges to their numerical simulation. Several successful schemes were developed so far and are mainly based on finite differences, finite volume and discontinuous Galerkin (dG) finite element schemes. For an overview on these schemes we refer to [GR96, Krö97, LeV02, Coc03, HW08] and their references. In this work we are interested in a posteriori error control of hyperbolic systems while solutions are still smooth. Our main tools are appropriate reconstructions of the discontinuous Galerkin schemes considered and relative entropy estimates.The first systematic a posteriori analysis for numerical approximations of scalar conservation laws accompanied with corresponding adaptive algorithms, can be traced back to [KO00, GM00], see also [Coc03,DMO07] and their references. These estimates were derived by employing Kruzkov's estimates. A posteriori results for systems were derived in [Laf08, Laf04] for front tracking and Glimm's schemes, see also [KLY10]. For recent a posteriori analysis for well balanced schemes for a damped semilinear wave equation we refer to [AG13].We aim at providing a rigorous a posteriori error estimate for semidiscrete dG schemes applied to systems of hyperbolic conservation laws which are of optimal order. The extension of these results to fully discrete schemes is obviously an important point but exceeds the scope of the work at hand. Our analysis is based on an extension of the reconstruction technique, developed mainly for discretisations of parabolic problems, see [Mak07] and references therein, to space discretisations in the hyperbolic setting. The main idea of the reconstruction technique is to introduce an intermediate function, which we will denote u, which solves a perturbed partial differential equation (PDE). This perturbed PDE is constructed in such a way that this u is sufficiently close to both the approximate solution, denoted u h and the exact solution to the conservation law, denoted u. Then, typically

Relative entropy based error estimates for discontinuous Galerkin schemes

“…Provided û x (s, ·) L ∞ (T) is uniformly bounded in h we expect the right hand side of (21) to be of optimal order. This is confirmed by numerical experiments, given in [9], in case of an at least Lipschitz continuous entropy solution.…”

“…At this point assumption (11) is used in order to obtain an error estimate of optimal order. For the technical details we refer to [9].…”

“…Let u be an entropy solution of (1) with periodic boundary conditions. Provided u and the reconstructionû of the numerical solution u h , given by (9), take values in some compact and convex K ⊂ U , then for any T > 0 there exists a computable constant C depending on K and T so that for 0 ≤ t ≤ T the error between u h and u satisfies (20).…”

“…Proofs of all assertions made in the first part of this note can be found in [9]. Let us give a short account on other a posteriori results for dG approximations of hyperbolic conservation laws.…”

Analysis of discontinuous Galerkin methods using mesh-dependent norms and applications to problems with rough data

“…The orthogonality condition follows from integrating both sides of (14) by parts. To see (16) we note that…”