Error analysis of a dynamic model adaptation procedure for nonlinear hyperbolic equations

Cancès, Clément; Coquel, Fré́dé́ric; Godlewski, Edwige; Mathis, Hélène; Seguin, Nicolas

doi:10.4310/cms.2016.v14.n1.a1

Cited by 4 publications

(8 citation statements)

References 32 publications

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“…If the homogeneous system (33) is solved by the fine numerical scheme (16)(17)(18)(19) and if the system of ODE's (34) is solved by the implicit Euler scheme (18)(19), then the steps (C'.a-C'.d) are equivalent to step C. This decomposition of the adaptive method enables us to state the following fundamental stability property:…”

Section: Alternative Formulation and Propertiesmentioning

confidence: 98%

“…Moreover, if the fine numerical scheme (16)(17)(18)(19) is entropy satisfying in the sense of Harten et al [41], then the fully discrete adaptive method (A-B-C) is entropy decreasing, that is to say…”

Section: Proposition 33 the Continuous-in-space Version Of Adaptive mentioning

confidence: 99%

“…Moreover, the authors with Frédéric Coquel propose in [16] a complete analysis for a simpler model to understand how to balance the errors due to the coupling of models with the errors due to the dynamic and local replacement of the fine model by the coarse model. This analysis enables to design a multiscale method with model adaptation.…”

Section: Application To Dynamic Model Adaptationmentioning

confidence: 99%

“…To do so, we first use the Chapman-Enskog expansion at the discrete level on the numerical scheme (16)(17)(18)(19). (16)(17)(18)(19).…”

Section: Numerical Indicatormentioning

confidence: 99%

“…System (29) is the continuous-in-space version of the fine numerical scheme (16)(17)(18)(19). Indeed, writing the Rankine-Hugoniot jump relations for (29) through {t = t n } yields…”

Section: Alternative Formulation and Propertiesmentioning

confidence: 99%

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Dynamic Model Adaptation for Multiscale Simulation of Hyperbolic Systems with Relaxation

et al. 2014

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In numerous industrial CFD applications, it is usual to use two (or more) different codes to solve a physical phenomenon: where the flow is a priori assumed to have a simple behavior, a code based on a coarse model is applied, while a code based on a fine model is used elsewhere. This leads to a complex coupling problem with fixed interfaces. The aim of the present work is to provide a numerical indicator to optimize to position of these coupling interfaces. In other words, thanks to this numerical indicator, one could verify if the use of the coarser model and of the resulting coupling does not introduce spurious effects. In order to validate this indicator, we use it in a dynamical multiscale method with moving coupling interfaces. The principle of this method is to use as much as possible a coarse model instead of the fine model in the computational domain, in order to obtain an accuracy which is comparable with the one provided by the fine model. We focus here on general hyperbolic systems with stiff relaxation source terms together with the corresponding 123 J Sci Comput hyperbolic equilibrium systems. Using a numerical Chapman-Enskog expansion and the distance to the equilibrium manifold, we construct the numerical indicator. Based on several works on the coupling of different hyperbolic models, an original numerical method of dynamic model adaptation is proposed. We prove that this multiscale method preserves invariant domains and that the entropy of the numerical solution decreases with respect to time. The reliability of the adaptation procedure is assessed on various 1D and 2D test cases coming from two-phase flow modeling.

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Section: Alternative Formulation and Propertiesmentioning

confidence: 98%

Section: Proposition 33 the Continuous-in-space Version Of Adaptive mentioning

confidence: 99%

Section: Application To Dynamic Model Adaptationmentioning

confidence: 99%

“…To do so, we first use the Chapman-Enskog expansion at the discrete level on the numerical scheme (16)(17)(18)(19). (16)(17)(18)(19).…”

Section: Numerical Indicatormentioning

confidence: 99%

“…System (29) is the continuous-in-space version of the fine numerical scheme (16)(17)(18)(19). Indeed, writing the Rankine-Hugoniot jump relations for (29) through {t = t n } yields…”

Section: Alternative Formulation and Propertiesmentioning

confidence: 99%

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Dynamic Model Adaptation for Multiscale Simulation of Hyperbolic Systems with Relaxation

et al. 2014

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Source Terms

Godlewski¹,

Raviart²

2020

Applied Mathematical Sciences

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Adjoint-Based Adaptive Model and Discretization for Hyperbolic Systems with Relaxation

Dronnier¹,

Renac²

2019

Multiscale Model. Simul.

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In this work, we use an adjoint-weighted residuals method for the derivation of an a posteriori model and discretization error estimators in the approximation of solutions to hyperbolic systems with stiff relaxation source terms and multiscale relaxation rates. These systems are parts of a hierarchy of models where the solution reaches different equilibrium states associated to different relaxation mechanisms. The discretization is based on a discontinuous Galerkin method which allows to account for the local regularity of the solution during the discretization adaptation. The error estimators are then used to design an adaptive model and discretization procedure which selects locally the model, the mesh, and the order of the approximation and balances both error components. Coupling conditions at interfaces between different models are imposed through local Riemann problems to ensure the transfer of information. The reliability of the present hpm-adaptation procedure is assessed on different test cases involving a Jin-Xin relaxation system with multiscale relaxation rates, and results are compared with standard hp-adaptation.

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Error analysis of a dynamic model adaptation procedure for nonlinear hyperbolic equations

Cited by 4 publications

References 32 publications

Dynamic Model Adaptation for Multiscale Simulation of Hyperbolic Systems with Relaxation

Dynamic Model Adaptation for Multiscale Simulation of Hyperbolic Systems with Relaxation

Source Terms

Adjoint-Based Adaptive Model and Discretization for Hyperbolic Systems with Relaxation

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