Seminar in Numerical Analysis: Maryna Kachanovska (ENSTA ParisTech)

Transparent boundary conditions for sound propagation in human lungs, modelled by fractal trees

In this work we consider the problem of the sound propagation in a bronchial network. Asymptotically, this phenomenon can be modelled by a weighted wave equation posed on a fractal (i.e. self-similar) 1D tree. The principal difficulty for the numerical resolution of the problem is the 'infiniteness' of the geometry. To deal with this issue, we will present transparent boundary conditions, used to truncate the computational domain to a finite subtree.

The construction of such transparent conditions relies on the approximation of the Dirichlet-to-Neumann (DtN) operator, whose symbol is a meromorphic function that satisfies a certain non-linear functional equation. We present two approaches to approximate the DtN in the time domain, alternative to the low-order absorbing boundary conditions, which appear inefficient in this case.

The first approach stems from the use of the convolution quadrature (cf. [Lubich 1988], [Banjai, Lubich, Sayas 2016]), which consists in constructing an exact DtN for a semi-discretized in time problem. In this case the combination of the explicit leapfrog method for the volumic terms and the implicit trapezoid rule for the boundary terms leads to a second-order scheme stable under the classical CFL condition.

The second approach is motivated by the Engquist-Majda ABCs (cf. [Engquist, Majda 1977]), and consists in approximating the DtN by local operators, obtained from the truncation of the meromorphic series which represents the symbol of the DtN. We show how the respective error can be controlled and provide some complexity estimates.

This is a joint work with Patrick Joly (INRIA, France) and Adrien Semin (TU Darmstadt, Germany).

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