Open Access
FALL 2014 Convolution spline approximations for time domain boundary integral equations
Penny J. Davies, Dugald B Duncan
J. Integral Equations Applications 26(3): 369-410 (FALL 2014). DOI: 10.1216/JIE-2014-26-3-369


We introduce a new ``convolution spline'' temporal approximation of time domain boundary integral equations (TDBIEs). It shares some properties of convolution quadrature (CQ) but, instead of being based on an underlying ODE solver, the approximation is explicitly constructed in terms of compactly supported basis functions. This results in sparse system matrices and makes it computationally more efficient than using the linear multistep version of CQ for TDBIE time-stepping. We use a Volterra integral equation (VIE) to illustrate the derivation of this new approach: at time step $t_n = n\dt$ the VIE solution is approximated in a backwards-in-time manner in terms of basis functions $\phi_j$ by $u(t_n-t) \approx \sum_{j=0}^n u_{n-j}\,\phi_j(t/\dt)$ for $t \in [0,t_n]$. We show that using isogeometric B-splines of degree $m\ge 1$ on $[0,\infty)$ in this framework gives a second order accurate scheme, but cubic splines with the parabolic runout conditions at $t=0$ are fourth order accurate. We establish a methodology for the stability analysis of VIEs and demonstrate that the new methods are stable for non-smooth kernels which are related to convergence analysis for TDBIEs, including the case of a Bessel function kernel oscillating at frequency $\oo(1/\dt)$. Numerical results for VIEs and for TDBIE problems on both open and closed surfaces confirm the theoretical predictions.


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Penny J. Davies. Dugald B Duncan. "Convolution spline approximations for time domain boundary integral equations." J. Integral Equations Applications 26 (3) 369 - 410, FALL 2014.


Published: FALL 2014
First available in Project Euclid: 31 October 2014

zbMATH: 1307.65127
MathSciNet: MR3273900
Digital Object Identifier: 10.1216/JIE-2014-26-3-369

Primary: 65M12 , 65R20

Keywords: Convolution quadrature , time dependent boundary integral equations , Volterra integral equations

Rights: Copyright © 2014 Rocky Mountain Mathematics Consortium

Vol.26 • No. 3 • FALL 2014
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