指数龙格-库塔方法的阶数降低：非交换算子的四阶格式
Order Reduction of Exponential Runge--Kutta Methods: Fourth-Order Schemes for Non-Commuting Operators
- Department of Mathematics and Statistics, University of Helsinki（赫尔辛基大学数学与统计系）
- Faculty of Mathematics and Physics, Charles University（查理大学数学物理学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文扩展了指数龙格-库塔方法的收敛性分析至四阶，针对非交换算子情形，通过误差递推和缺陷框架识别了阶数降低项，数值实验验证了理论预测。
AI中文摘要：
本文将对线性抛物问题 $u'(t) + Au(t) = Bu(t)$ 的显式指数龙格-库塔方法的收敛性分析从三阶情形扩展到四阶格式，其中 $A$ 生成解析半群，$B$ 关于 $A$ 相对有界。通过建立全局误差递推关系并扩展基于缺陷的分析框架，我们识别了当 $A$ 和 $B$ 不交换时导致刚性阶数降低的项。数值实验在一个非交换的对流-扩散问题上进行，以验证理论结果。使用 Krogstad 和 Strehmel & Weiner 的经典四阶段四阶格式进行的数值测试显示出约 2.75 的观测收敛阶，这与收敛性分析的理论预测相符。
英文摘要：
This paper extends the convergence analysis of explicit exponential Runge--Kutta methods for linear parabolic problems $u'(t) + Au(t) = Bu(t)$, where $A$ generates an analytic semigroup and $B$ is relatively bounded with respect to $A$, from the third-order case to fourth-order schemes. By establishing the global error recursion relation and extending the defect-based analytical framework, we identify the terms responsible for stiff order reduction when $A$ and $B$ do not commute. Numerical experiments are performed on a non-commuting advection-diffusion problem to validate the theoretical results. Numerical tests using the classical four-stage, fourth-order schemes of Krogstad and Strehmel \& Weiner exhibit an observed convergence order of approximately 2.75, which matches the theoretical prediction from the convergence analysis.
