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Diffusive Approximation of a Time-Fractional Burger's Equation in Nonlinear Acoustics

Lombard, Bruno and Matignon, Denis Diffusive Approximation of a Time-Fractional Burger's Equation in Nonlinear Acoustics. (2016) SIAM Journal on Applied Mathematics, 76 (5). 1765-1791. ISSN 0036-1399

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Official URL: http://dx.doi.org/10.1137/16M1062491

Abstract

A fractional time derivative is introduced into Burger's equation to model losses of nonlinear waves. This term amounts to a time convolution product, which greatly penalizes the numerical modeling. A diffusive representation of the fractional derivative is adopted here, replacing this nonlocal operator by a continuum of memory variables that satisfy local-in-time ordinary differential equations. Then a quadrature formula yields a system of local partial differential equations, well-suited to numerical integration. The determination of the quadrature coefficients is crucial to ensure both the well-posedness of the system and the computational efficiency of the diffusive approximation. For this purpose, optimization with constraint is shown to be a very efficient strategy. Strang splitting is used to solve successively the hyperbolic part by a shock-capturing scheme and the diffusive part exactly. Numerical experiments are proposed to assess the efficiency of the numerical modeling and to illustrate the effect of the fractional attenuation on the wave propagation.

Item Type:Article
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:Other partners > Aix-Marseille Université - AMU (FRANCE)
French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Université de Toulouse > Institut Supérieur de l'Aéronautique et de l'Espace - ISAE-SUPAERO (FRANCE)
Other partners > Université de Provence - Aix-Marseille I (FRANCE)
Other partners > Ecole Centrale Marseille (FRANCE)
Other partners > Université de la Méditerranée - Aix-Marseille II (FRANCE)
Laboratory name:
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Deposited By: Denis Matignon
Deposited On:10 Oct 2016 12:16

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