Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics

Bruno Lombard 1 Denis Matignon 2
1 O&I - Ondes et Imagerie
LMA - Laboratoire de Mécanique et d'Acoustique [Marseille]
2 Univ. Toulouse, ISAE-Supaéro
ISAE-SUPAERO - Institut Supérieur de l'Aéronautique et de l'Espace
Abstract : A fractional time derivative is introduced into the 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.
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  • HAL Id : hal-01276821, version 3
  • ARXIV : 1602.07205

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Bruno Lombard, Denis Matignon. Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics. SIAM Journal on Applied Mathematics, Society for Industrial and Applied Mathematics, 2016, 76 (6), pp.1765-1791. ⟨hal-01276821v3⟩

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