Stability and Convergence of a Class of Finite Element Schemes for Hyperbolic 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.Date: September 7, 2017. JG partially supported by the German Research Foundation (DFG) via SFB TRR 75 'Tropfendynamische Prozesse unter extremen Umgebungsbedingungen'.

Section: Introductionmentioning

confidence: 99%

A Posteriori Analysis of Fully Discrete Method of Lines Discontinuous Galerkin Schemes for Systems of Conservation Laws

Dedner¹,

Giesselmann²

2016

“…Secondly, the relative entropy is an L 2 framework, thus, the residuals in the perturbed equation satisfied byû need to be in L 2 . Relative entropy techniques for the a priori error analysis of approximations of systems of conservation laws were first used in [AMT04]. For other works concerning analysis of schemes for systems of conservation laws see, e.g.…”

mentioning

confidence: 99%

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

Giesselmann¹,

Makridakis²,

Pryer³

2015

Self Cite

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

“…Related earlier work established convergence of the similar shock-capturing streamline diffusion finite element method (see [24], [25]), which, however, is not a DG method, and error estimates for this method were also given in [3]. The only work that we know of that proves convergence of any DG method for any system of conservation laws is a very recent paper of Arvanitis, Makridakis, and Tzavaras [1] that considers a DG method based on a relaxation approximation. This method, which does not reduce to (1.3) when k = 0, is shown in [1] to converge for systems that can be controlled using compensated compactness techniques.…”

Section: Introductionmentioning

confidence: 99%

“…The only work that we know of that proves convergence of any DG method for any system of conservation laws is a very recent paper of Arvanitis, Makridakis, and Tzavaras [1] that considers a DG method based on a relaxation approximation. This method, which does not reduce to (1.3) when k = 0, is shown in [1] to converge for systems that can be controlled using compensated compactness techniques.…”

Section: Introductionmentioning

confidence: 99%

“…Indeed, at the end of section 6 we show by example that the solution of scheme (1.3) need not be unique without suitable restrictions on the mesh, even in the scalar case. 1 It should be noted that, despite these theoretical difficulties, in practice approximate solutions can be computed quite efficiently using Newton's method; see [20] and [18] for related numerical results.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Convergence of an Implicit Spacetime Godunov Finite Volume Method for a Class of Hyperbolic Systems

Jegdic¹,

Jerrard²

2006