High-order contour-integral methods for strongly continuous semigroups

Abstract

Exponential integrators based on contour integral representations lead to powerful numerical solvers for a variety of ODEs, PDEs, and other time-evolution equations. They are embarrassingly parallelizable and can lead to fast and highly accurate global-in-time algorithms. However, core theoretical challenges restrict their applicability to evolution equations governed by analytic semigroups, which have a strong smoothing property, as in strongly damped wave propagation and diffusive transport. In this talk, we explain how carefully regularized contour integral representations can be used to construct high-order global-in-time approximations for the much larger class of strongly continuous semigroups, which includes a wide range of non-smoothing wave- and advection-dominated phenomena. Our algorithms are accompanied by explicit high-order error bounds and near-optimal parameter selection, and we demonstrate key features of the approximations with examples drawn from advection-dominated flows.