## Journal of Integral Equations and Applications

- J. Integral Equations Applications
- Volume 26, Number 3 (2014), 369-410.

### Convolution spline approximations for time domain boundary integral equations

Penny J. Davies and Dugald B Duncan

#### Abstract

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.

#### Article information

**Source**

J. Integral Equations Applications, Volume 26, Number 3 (2014), 369-410.

**Dates**

First available in Project Euclid: 31 October 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.jiea/1414761103

**Digital Object Identifier**

doi:10.1216/JIE-2014-26-3-369

**Mathematical Reviews number (MathSciNet)**

MR3273900

**Zentralblatt MATH identifier**

1307.65127

**Subjects**

Primary: 65R20: Integral equations 65M12: Stability and convergence of numerical methods

**Keywords**

Convolution quadrature Volterra integral equations time dependent boundary integral equations

#### Citation

Davies, Penny J.; Duncan, Dugald B. Convolution spline approximations for time domain boundary integral equations. J. Integral Equations Applications 26 (2014), no. 3, 369--410. doi:10.1216/JIE-2014-26-3-369. https://projecteuclid.org/euclid.jiea/1414761103