On the Convergence and Numerical Stability of the Second Waterman Scheme for Approximation of the Acoustic Field Scattered by a Hard Object

Dallas, A.G.
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Department of Mathematical Sciences
The numerical schemes of P.C. Waterman (J. Acoust. Soc. Am.45 (1969), 1417-1429), frequently referred to under the name of "the T-Matrix method," have formed the basis for many scattering computations in many settings. However, no successful analyses of the algorithms have been published, so the limitations on their range of applicability and numerical stability remain largely unknown; this is of particular importance because of the apparently inconsistent success achieved in numerical experiments. Here, we give an operator condition that guarantees the viability of the algorithm and mean-square convergence of the far-field patterns generated by the second Waterman scheme for the case of time-harmonic acoustic scattering by a hard obstacle; we prove further that the operator condition holds at least whenever the scattering obstacle is ellipsoidal. For the convergence proof, we also assume that the square of the wavenumber is not an interior Dirichlet eigenvalue for the negative Laplacianl in the contrary case, we show that the algorithm is at best numerically ill-coordinated. With this and previous experience in numerical applications, it appears that the performance of the algorithm is markedly shape-dependent; for certain obstacles, e.g., ellipsoids, instabilities are so localized in wavenumber that they are practically numerically irrelevant, while it is not clear whether the erratic results found in applications to various other shapes arise from a failure of convergence or form numerical instability.