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Stability and convergence of collocation schemes for retarded potential integral equations

Davies, P.J. and Duncan, D.B. (2004) Stability and convergence of collocation schemes for retarded potential integral equations. SIAM Journal on Numerical Analysis, 42 (3). pp. 1167-1188. ISSN 0036-1429

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Abstract

Time domain boundary integral formulations of transient scattering problems involve retarded potential integral equations. Solving such equations numerically is both complicated and computationally intensive, and numerical methods often prove to be unstable. Collocation schemes are easier to implement than full finite element formulations, but little appears to be known about their stability and convergence. Here we derive and analyze some new stable collocation schemes for the single layer equation for transient acoustic scattering, and use (spatial) Fourier and (temporal) Laplace transform techniques to demonstrate that such stable schemes are second order convergent.