Maximal regularity for time-stepping schemes arising from convolution quadrature of non-local in time equations

Abstract

We study discrete time maximal regularity in Lebesgue spaces of sequences for time-stepping schemes arising from Lubich’s convolution quadrature method. We show minimal properties on the quadrature weights that determines a wide class of implicit schemes. For an appropriate choice of the weights, we are able to identify the θ-method as well as the backward differentiation formulas and the L1-scheme. Fractional versions of these schemes, some of them completely new, are also shown, as well as their representation by means of the Gr¨unwald–Letnikov fractional order derivative. Our results extend and improve some recent results on the subject and provide new insights on the basic nature of the weights that ensure maximal regularity.