Continuous dependence and error estimation for viscosity methods

Cockburn, Bernardo

doi:10.1017/s0962492902000107

Cited by 26 publications

(21 citation statements)

References 89 publications

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“…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.…”

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confidence: 99%

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

Giesselmann¹,

Makridakis²,

Pryer³

2015

SIAM J. Numer. Anal.

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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

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confidence: 99%

“…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.…”

mentioning

confidence: 99%

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

Giesselmann¹,

Makridakis²,

Pryer³

2015

SIAM J. Numer. Anal.

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“…We note that the arguments and the usages of the PDE spectral estimate are quite different in [285] and in [212,213]. A topological argument was used in [285], in which the PDE spectral estimate is embedded, to obtain a fine a posteriori error bound; while a sharp continuous dependence estimate for the PDE solution was used, in which the PDE spectral estimate was crucially utilized, to derive fine a posteriori error estimates in [212,213] (also see [114]). Because this continuous dependence argument is very simple and may be applicable to other evolution PDEs, we briefly explain it below.…”

Section: Coarse and Fine A Posteriori Errormentioning

confidence: 99%

The phase field method for geometric moving interfaces and their numerical approximations

Feng

2020

Handbook of Numerical Analysis

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This chapter surveys recent numerical advances in the phase field method for geometric surface evolution and related geometric nonlinear partial differential equations (PDEs). Instead of describing technical details of various numerical methods and their analyses, the chapter presents a holistic overview about the main ideas of phase field modeling, its mathematical foundation, and relationships between the phase field formalism and other mathematical formalisms for geometric moving interface problems, as well as the current state-of-the-art of numerical approximations of various phase field models with an emphasis on discussing the main ideas of numerical analysis techniques. The chapter also reviews recent development on adaptive grid methods and various applications of the phase field modeling and their numerical methods in materials science, fluid mechanics, biology and image science. Key words and phrases. Phase field method, geometric law, curvature-driven flow, geometric nonlinear PDEs, finite difference methods, finite element methods, spectral methods, discontinuous Galerkin methods, adaptivity, coarse and fine error estimates, convergence of numerical interfaces, nonlocal and stochastic phase field models, microstructure evolution, biology and image science applications.

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“…The extension of a posteriori error estimators to first order PDEs is far from trivial. A limited number of attempts were done to obtain a posteriori estimators for first order hyperbolic PDEs, either for the pure advection equation [30], linear symmetric hyperbolic systems [29,49], nonlinear scalar hyperbolic equations [13,14], 1D nonlinear hyperbolic systems [31] and multi-dimensional nonlinear hyperbolic systems [32]. An estimator was proposed and tested for the linearized Euler equations for compressible flows [46].…”

Section: Introductionmentioning

confidence: 99%

On the use of anisotropic a posteriori error estimators for the adaptative solution of 3D inviscid compressible flows

Bourgault

Picasso

Alauzet

et al. 2008

Numerical Methods in Fluids

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SUMMARYThis paper describes the use of an a posteriori error estimator to control anisotropic mesh adaptation for computing inviscid compressible flows. The a posteriori error estimator and the coupling strategy with an anisotropic remesher are first introduced. The mesh adaptation is controlled by a single parameter T OL in regions where the solution is regular, while a condition on the minimal element size hmin is enforced across solution discontinuities. This hmin condition is justified on the basis of an asymptotic analysis. The efficiency of the approach is tested with a supersonic flow over an aircraft. The evolution of a mesh adaptation/flow solution loop is shown, together with the influence of the parameters T OL and hmin. We verify numerically that the effect of varying hmin is concordant with the conclusions of the asymptotic analysis, giving hints on the selection of hmin with respect to T OL. Finally we check that the results obtained with the a posteriori error estimator are at least as accurate as those obtained with anisotropic a priori error estimators. All the results presented can be obtained using a standard desktop computer, showing the efficiency of these adaptative methods. Copyright

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Continuous dependence and error estimation for viscosity methods

Cited by 26 publications

References 89 publications

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

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

The phase field method for geometric moving interfaces and their numerical approximations

On the use of anisotropic a posteriori error estimators for the adaptative solution of 3D inviscid compressible flows

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