High order surface radiation conditions for time-harmonic waves in exterior domains

Abstract

We formulate a new family of high order on-surface radiation conditions to approximate the outgoing solution to the Helmholtz equation in exterior domains. Motivated by the pseudo-differential expansion of the Dirichlet-to-Neumann operator developed by Antoine et al. (J. Math. Anal. Appl. 229:184-211, 1999), we design a systematic procedure to apply pseudo-differential symbols of arbitrarily high order. Numerical results are presented to illustrate the performance of the proposed method for solving both the Dirichlet and the Neumann boundary value problems. Possible improvements and extensions are also discussed.

Corner treatments for high-order local absorbing boundary conditions in high-frequency acoustic scattering

“…Local boundary conditions can be derived by approximating the square root in the symbol of the DtN operator (1). In their seminal paper, Engquist and Majda [31] derived a family of local boundary conditions by using a Padé approximation of the square root.…”

Well-Posedness for Photoacoustic Tomography with Fabry--Perot Sensors

“…The subject of nonreflecting or absorbing boundary conditions is beyond the scope of this paper. We refer to [9,13,6,20,1] for some relevant articles on that topic. We consider…”

On-surface radiation condition for multiple scattering of waves

“…Figure 1 displays (a) the sparsity pattern of the matrix A 1,2 for the configuration of two spheres from subsection 6.1, (b) the sparsity pattern of the discrete Laplace-Beltrami operator ∆ T 1 , and (c) CPU time for the application of the propagator (30) as a function of DOF. For the numerical results presented in the next section, we apply a fixed-point iteration to solve the system (32). Due to the convexity of the surfaces ∂Ω j for all j = 1, 2, ..., J, the norm of the propagation operator P i, j decreases as the distance between the i th and j th obstacles increases.…”

“…Possible future work includes the extension to electromagnetic and elastic waves that govern important engineering applications. It is also possible to increase the order of approximations of the Dirichlet-to-Neumann map to improve the accuracy of the OSRC [32]. Here we have considered the second order differential approximation (16).…”

“…Other improvements have been accomplished by Jones [11,12,13], Ammari [14,15], Calvo et al [16,17], Barucq et al [18,19,20], Chaillat et al [21,22] and Darbas et al [2,3,4,5,23]. See also [24,25,26,27,28,29,30,31,25,32].…”