Abstract
We derive a new \convolution spline" approximation method for convolution Volterra integral equations. This shares some properties of convolution quadrature, but instead of being based on an underlying ODE solver is explicitly constructed in terms of basis functions which have compact support. At time step tn = nh > 0, the solution is approximated in a \backward time" manner in terms of basis functions j by u(tn􀀀t) Pnj=0 un􀀀jj(t=h) for t 2 [0; tn]. We carry out a detailed analysis for Bspline basis functions, but note that the framework is more general than this. For Bsplines of degree m 1 we show that the schemes converge at the rate O(h2) when the kernel is
suciently smooth. We also establish a methodology for their stability analysis and obtain new stability results for several nonsmooth kernels, including the case of a highly oscillatory Bessel function kernel (in which the oscillation frequency can be O(1=h)). This is related to convergence analysis for approximation of time domain boundary integral equations (TDBIEs), and provides evidence that the new convolution spline approach could provide a useful timestepping mechanism for TDBIE problems. In particular, using
compactly supported basis functions would give sparse system matrices.
suciently smooth. We also establish a methodology for their stability analysis and obtain new stability results for several nonsmooth kernels, including the case of a highly oscillatory Bessel function kernel (in which the oscillation frequency can be O(1=h)). This is related to convergence analysis for approximation of time domain boundary integral equations (TDBIEs), and provides evidence that the new convolution spline approach could provide a useful timestepping mechanism for TDBIE problems. In particular, using
compactly supported basis functions would give sparse system matrices.
Original language  English 

Pages (fromto)  369410 
Number of pages  42 
Journal  Journal of Integral Equations and Applications 
Volume  26 
Issue number  3 
DOIs  
Publication status  Published  31 Oct 2014 
Keywords
 Convolution quadrature
 Volterra integral equations
 time dependent boundary integral equations
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Dugald Black Duncan
 Research Centres and Themes, Energy Academy  Professor
 School of Mathematical & Computer Sciences  Professor
 School of Mathematical & Computer Sciences, Mathematics  Professor
Person: Academic (Research & Teaching)