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

Abstract : We propose a dynamic model adaptation method for a nonlinear conservation law coupled with an ordinary differential equation. This model, called the ''fine model", involves a small time scale and setting this time scale to 0 leads to a classical conservation law, called the ''coarse model", with a flux which depends on the unknown and on space and time. The dynamic model adaptation consists in detecting the regions where the fine model can be replaced by the coarse one in an automatic way, without deteriorating the accuracy of the result. To do so, we provide an error estimate between the solution of the fine model and the solution of the adaptive method, enabling a sharp control of the different parameters. This estimate rests upon stability results for conservation laws with respect to the flux function. Numerical results are presented at the end and show that our estimate is optimal.
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Communications in Mathematical Sciences, International Press, 2016, 14 (1), pp.1-30
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Clément Cancès, Frédéric Coquel, Edwige Godlewski, Hélène Mathis, Nicolas Seguin. Error analysis of a dynamic model adaptation procedure for nonlinear hyperbolic equations. Communications in Mathematical Sciences, International Press, 2016, 14 (1), pp.1-30. 〈hal-00852101〉

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