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

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.

Asymptotic stability of the multidimensional wave equation coupled with classes of positive-real impedance boundary conditions

“…The motivation behind the definition of this kernel is physical as it models passive systems that arise in e.g. electromagnetics [21], viscoelasticity [17,41], and acoustics [28,37,48]. By assumption, the right-hand side of (4) is a sum of positive-real kernels that each admit a dissipative realization.…”

“…The typical extended diffusive operator is the Riemman-Liouville fractional derivative [55, § 2.3] [43], obtained forẑ(s) = s 1−α and dµ given by (35), which satisfies the condition (55). For this measure dµ, choosing the initialization ϕ(0, ξ) = u(0) /ξ in (56) yields the Caputo derivative [37].…”

Energy analysis and discretization of nonlinear impedance boundary conditions for the time-domain linearized Euler equations

“…(As a matter of fact, this is also the case for the admittance ŷ phys .) Since the computations are similar to that carried out in [5], only the key steps are provided; the reader is referred to [5,58] and references therein for background on oscillatory-diffusive representations.…”

A two-way model for nonlinear acoustic waves in a non-uniform lattice of Helmholtz resonators

“…Contrary to [28], all the initial conditions are null: u ± = 0, p = 0, ∂p ∂t = 0. If this were not the case, then the diffusive representation of the 3/2 derivative would imply non-null initial conditions on ξ in (30). The interested reader is referred to [30] for additional details on this topic.…”