Transparent boundary conditions for wave propagation in fractal trees: convolution quadrature approach

• Published in: Numerische Mathematik Vol. 146; no. 2; pp. 281 - 334
• Main Authors: Joly, Patrick, Kachanovska, Maryna
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.10.2020; Springer Nature B.V; Springer Verlag
• Subjects: Algorithms; Boundary conditions; Convolution; Fractals; Mathematical analysis; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Numerical Analysis; Numerical and Computational Physics; Quadratures; Simulation; Sound propagation; Stability analysis; Theoretical; Wave equations; Wave propagation; 65M60; 35R02; 65M12; 65M06; Convolution quadrature; fractal trees; transparent boundary conditions; quantum graph; DtN operator
• ISSN: 0029-599X; 0945-3245
• DOI: 10.1007/s00211-020-01145-9

In this work we propose high-order transparent boundary conditions for the weighted wave equation on a fractal tree, with an application to the modeling of sound propagation in a human lung. This article follows the recent work (Joly et al. in Netw Heterog Media 14(2):205–264, 2019), dedicated to the mathematical analysis of the corresponding problem and the construction of low-order absorbing boundary conditions. The method proposed in this article consists in constructing the exact (transparent) boundary conditions for the semi-discretized problem, in the spirit of the convolution quadrature method developed by Ch. Lubich. We analyze the stability and convergence of the method, and propose an efficient algorithm for its implementation. The exposition is concluded with numerical experiments.