发表机构
Lawrence Livermore National Laboratory; Indian Institute of Technology Kanpur; New Jersey Institute of Technology; Temple University（劳伦斯利弗莫尔国家实验室; 坎普尔印度理工学院; 新泽西理工学院; 天普大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文利用先前推导的锐利阶条件，构造了首批无阶数降低的四阶和五阶对角隐式龙格-库塔方法，数值实验表明其在刚性半线性问题上优于经典方法。
AI 中文摘要
对角隐式龙格-库塔（DIRK）方法是求解刚性常微分方程（ODE）系统的一类重要数值方法。刚性不仅对龙格-库塔方法构成稳定性挑战，还可能降低收敛阶数。当经典收敛分析所用假设（如步长渐近趋于零）不成立时，便会出现所谓的阶数降低现象。在作者先前的一篇论文中，针对一大类半线性常微分方程，推导了龙格-库塔方法的锐利阶条件与全局误差界，这些条件与误差界关于刚性是一致成立的。在本工作中，利用这些条件构造了首批满足这些条件的四阶和五阶DIRK方法，从而不出现阶数降低。数值结果表明，对于一大类相关的非线性测试问题，这些新方法能有效缓解阶数降低，通过嵌入格式准确估计局部误差以用于自适应步长控制，并且可以优于经典DIRK方法。
英文摘要
Diagonally implicit Runge-Kutta (DIRK) methods are a prominent class of numerical methods for solving stiff systems of ordinary differential equations (ODEs). Stiffness does not only impose stability challenges on Runge-Kutta methods; it can also degrade the order of convergence. This so-called order reduction phenomenon occurs when assumptions used for classical convergence analysis, e.g., an asymptotically small step size, fail to hold. In a prior paper by the authors, sharp order conditions and global error bounds for Runge-Kutta methods were developed, which hold uniformly with respect to stiffness when applied to a wide class of semilinear ODEs. In this work, those conditions are leveraged to construct the first DIRK methods of order four and five which satisfy these conditions and thus do not exhibit order reduction. Numerical results demonstrate that for a broad class of relevant nonlinear test problems, these new methods successfully mitigate order reduction, accurately estimate local error via an embedding for adaptive step size control, and can outperform classical DIRK methods.
