Issue |
ESAIM: M2AN
Volume 59, Number 4, July-August 2025
|
|
---|---|---|
Page(s) | 2279 - 2304 | |
DOI | https://doi.org/10.1051/m2an/2025059 | |
Published online | 31 July 2025 |
Existence, uniqueness, and numerical solutions of the nonlinear periodic westervelt equation
1
Department of Mathematics, University of Klagenfurt, Carinthia, Austria
2
Austrian Institute of Technology GmbH, Vienna, Austria
* Corresponding author: benjamin.rainer@ait.ac.at
Received:
7
February
2025
Accepted:
23
June
2025
In this paper, we study the nonlinear periodic Westervelt equation with excitations located within a bounded domain in Rd, where d ∈ {2, 3}, subject to Robin boundary conditions. This problem is of particular interest for advancing imaging techniques that exploit nonlinearity of the acoustic propagation. We establish the existence and uniqueness of solutions in both the linear and the nonlinear setting, thereby allowing for spatially varying coefficients as relevant in quantitative imaging. Derivation of a multiharmonic formulation enables us to show the generation of higher harmonics (that is, responses at multiples of the fundamental frequency) due nonlinear wave propagation. An iterative scheme for solving the resulting system is proposed that relies on successive resolution of these higher harmonics, and its convergence under smallness conditions on the excitation is proven. Furthermore, we investigate the numerical solution of the resulting system of Helmholtz equations, employing a conforming finite element method for its discretization. Through an implementation of the proposed methodology, we illustrate how acoustic waves propagate in nonlinear media. This study aims to enhance our understanding of ultrasound propagation dynamics, which is essential for obtaining high-quality images from limited in vivo and boundary measurements.
Mathematics Subject Classification: 35A01 / 35A02 / 35G20 / 35A35
Key words: Nonlinear periodic Westervelt equation / well-posedness / numerical approximation
© The authors. Published by EDP Sciences, SMAI 2025
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