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A fractional Burgers equation arising in nonlinear acoustics: theory and numerics

Lombard, Bruno and Matignon, Denis and Le Gorrec, Yann A fractional Burgers equation arising in nonlinear acoustics: theory and numerics. (2013) In: NOLCOS - 9th IFAC Symposium on Nonlinear Control Systems, 4 September 2013 - 6 September 2013 (Toulouse, France).

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Official URL: http://www.ifac-papersonline.net/Detailed/62063.html

Abstract

The study of a fractional Burgers equation arising in nonlinear acoustics is presented. The motivation comes from an elementary model of shock waves in brass wind instruments, that is useful in musical acoustics. Such a model results from the coupling of a conservative nonlinear system with a dissipative term; here the dissipation is represented by a fractional derivative in time, for which equivalent diffusive representations can be efficiently used: an existence and uniqueness theorem for strong solutions is recalled in the first part. In a second part, ad hoc numerical schemes are derived, in order to capture all the physical phenomenon at stake in the original model, and to get rid, as far as possible, of the spurious numerical effects which are highly undesirable: conservative schemes for hyperbolic conservation laws, diffusive realizations for the fractional derivatives and integrals, and splitting to couple both aspects. Two interesting questions are still left open: recasting the coupled model in the setting of infinite-dimensional port-Hamiltonian systems (pHs) including dissipation, and proving existence of weak solutions to the original model, e.g. in the case of discontinuous inital data.

Item Type:Conference or Workshop Item (Paper)
Audience (conference):International conference proceedings
Uncontrolled Keywords:
Institution:French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Other partners > Ecole Nationale Supérieure de Mécanique et des Microtechniques - ENSMM (FRANCE)
Université de Toulouse > Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE)
Other partners > Université de Franche-Comté (FRANCE)
Other partners > Université de Provence - Aix-Marseille I (FRANCE)
Other partners > Université de Technologie de Belfort-Montbéliard - UTBM (FRANCE)
Other partners > Ecole Centrale Marseille (FRANCE)
Other partners > Institut des Sciences de l’Ingénierie et des Systèmes - INSIS (FRANCE)
Other partners > Université de la Méditerranée - Aix-Marseille II (FRANCE)
Other partners > Université Paul Cézanne - Aix-Marseille III (FRANCE)
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Deposited By: Denis Matignon
Deposited On:06 May 2013 13:10

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