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.

翻译：基于轮廓积分表示的指数积分器为各类常微分方程、偏微分方程及其他时间演化方程提供了强大的数值求解器。这些方法具有天然的并行性，并能生成全局时间近似解，可在有限时间范围内高效计算任意时刻的数值。然而，其核心理论局限将适用范围限制于解析半群（如抛物型方程）。本文通过精心正则化的轮廓积分表示，为更广泛且正则性较弱的强连续半群构造了一系列新型高阶求积格式。所提算法均配有显式的高阶误差界与接近最优的参数选择策略。我们通过Koopman算子理论中的奇异一阶偏微分方程展示了该系列格式的核心特性。