Convolution quadrature revisited (Q1770923)

This paper is a review on convolution quadratures. In fact they have many attractive features that let them work well in a lot of applications where the model equations are derived via a frequency-domain fundamental solution of the transfer equation, whereas the time-domain fundamental solution or impulse response is not available. Moreover such convolution quadrature methods enjoy excellent stability properties when used to discretize integral, integro-differential equations or integro-partial differential equations, etc. They are also useful as a theoretical tool in analyzing standard numerical time discretization methods through discretized operational calculus. In particular this paper closes a gap that was left in the theory of \textit{C. Lubich} [Numer. Math. 52, 129--145 (1988; Zbl 0637.65016)], by extending the known approximation results for the case of sectorial Laplace transforms to finite-part convolutions with nonintegrable kernel. The author also presents two new detailed and unified proofs of the optimal error bounds for both locally integrable and nonintegrable convolution kernels.