An efficient high-order Nyström scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

Abstract

This text proposes a fast, rapidly convergent Nyström method for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by inhomogeneous obstacles, while allowing the material properties to jump across the interface. The method works with overlapping coordinate charts as a description of the given scatterer. In particular, it employs "partitions of unity" to simplify the implementation of highorder quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations. To deal with the discontinuous material interface in a high-order manner, a specialized quadrature is used in the boundary region. The approach further utilizes an FFT based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of O(N log N ) for an N -point discretization. A detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance in terms of both speed and accuracy are presented in this paper. arXiv:1509.05063v1 [math.NA] 16 Sep 2015

Improved convergence of fast integral equation solvers for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

“…In recent years, a number of algorithms, including direct and iterative solvers, have been proposed for the solution of Lippmann-Schwinger equation. While we do not review all such contributions, some recent numerical methods include [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. Most fast algorithm, among the cited methods, while converging rapidly for smooth scattering media, yield only linear convergence in the presence of discontinuous scattering media.…”

“…and then using a high-order composite Newton-Cotes quadrature for the approximation of each of the integrals. However, as explained in [22], this strategy requires that we know I k s,t at off-grid points, particularly near t = 0 and t = 1. Clearly, the direct interpolation of I k s,t is not high-order accurate in view of the corner singularity at t = t. Again, following the strategy in [22], we resolve this by adding 2 × (Q − 1) × (Q − 1) additional grid, (Q − 1) × (Q − 1) in the vicinity of zero and (Q − 1) × (Q − 1) in the vicinity of one, where Q is the order of Newton-Cotes quadrature rule.…”

“…However, as explained in [22], this strategy requires that we know I k s,t at off-grid points, particularly near t = 0 and t = 1. Clearly, the direct interpolation of I k s,t is not high-order accurate in view of the corner singularity at t = t. Again, following the strategy in [22], we resolve this by adding 2 × (Q − 1) × (Q − 1) additional grid, (Q − 1) × (Q − 1) in the vicinity of zero and (Q − 1) × (Q − 1) in the vicinity of one, where Q is the order of Newton-Cotes quadrature rule. The values of I k s,t at these additional grid points are obtained by, first interpolating smooth density ϕ(τ, t ) at these extra grid points followed by integration in (29).…”

“…In addition, we also note that while the sources range in Ω B,h , the convolution needs to be evaluated not just on points in Ω B,h , but also on other grid points coming from Ω h \ Ω B that we refer to as "virtual sources". These additional sources, obviously, do not contribute to the sum in (30) but are included in the computation with "zero" strength (weight) so that the discrete convolutions get evaluated at the target points x q ∈ Ω h \ Ω B in an efficient manner using our FFT based acceleration strategy [22,38] that we explain next for completeness. The acceleration strategy seeks to compute a discrete convolution by substituting true sources by a certain set of "equivalent sources" located at a regular Cartesian grid on the two parallel faces of cell c i j (see Figure 5).…”

“…In addition to a detailed description of these methodologies, we also present a comparative performance study of the improved versions of these two and the Nyström solver in [J. Comput. Phys., 311 (2016), 258-274] through a wide range of numerical experiments.designed for a high-order treatment of discontinuous material interfaces [22] does converge rapidly with optimal computational cost, the griding strategy used therein, in certain cases, can restrict the methodology from achieving the theoretical computational complexity. Indeed, as we show later in this text, the scheme in [22] produces far less accurate approximations to the solution when compared to their counterparts obtained through the proposed approach.Apart from these, a couple of fast direct solvers with computational cost O(N 3/2 ) have also been proposed in [20,21] that primarily rely on quad-tree data structure.…”