A family of high-order accurate contour integral methods for strongly continuous semigroups

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 lead to global-in-time approximations that can be efficiently evaluated anywhere within a finite time horizon. However, there are core theoretical challenges that restrict their use cases to analytic semigroups, e.g., parabolic equations. In this article, we use carefully regularized contour integral representations to construct a family of new high-order quadrature schemes for the larger, less regular, class of strongly continuous semigroups. Our algorithms are accompanied by explicit high-order error bounds and near-optimal parameter selection. We demonstrate key features of the schemes on singular first-order PDEs from Koopman operator theory.