Communications in Mathematical Sciences

Volume 14 (2016)

Number 1

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

Pages: 1 – 30

DOI: https://dx.doi.org/10.4310/CMS.2016.v14.n1.a1

Authors

Clément Cancès (Laboratoire Jacques-Louis Lion, Sorbonne Universités, UPMC Univ Paris 6, Paris, France)

Frédéric Coquel (CNRS, UMR 7641, CMAP Ecole Polytechnique, Palaiseau, France)

Edwige Godlewski (Laboratoire Jacques-Louis Lion, Sorbonne Universités, UPMC Univ Paris 6, Paris, France)

Hélène Mathis (Laboratoire de Mathématiques Jean Leray, Université de Nantes, France)

Nicolas Seguin (Laboratoire Jacques-Louis Lion, Sorbonne Universités, UPMC Univ Paris 6, Paris, France)

Abstract

In order to theoretically validate a dynamic model adaptation method, we consider a simple case where the model error can be thoroughly analyzed. The dynamic model adaptation consists of detecting at each time step the region where a given fine model can be replaced by a corresponding coarse model in an automatic way, without deteriorating the accuracy of the result, and to couple the two models, each being computed on its respective region. Our fine model is a $2 \times 2$ system which involves a small time scale; setting this time scale to $0$ leads to a classical conservation law, the coarse model, with a flux which depends on the unknown and on space and time. The adaptation method provides an intermediate adapted solution which results from the coupling of both models at each time step. In order to obtain sharp and rigorous error estimates for the model adaptation procedure, a simple fine model is investigated, and smooth transitions between fine and coarse models are considered. We refine existing stability results for conservation laws with respect to the flux function which enables us to know how to balance the time step, the threshold for the domain decomposition, and the size of the transition zone. Numerical results are presented at the end and show that our estimate is optimal.

Keywords

conservation laws, error estimate, model adaptation, thick coupling interface

2010 Mathematics Subject Classification

35A35, 35B30, 35B45, 35L65

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Published 16 September 2015