用GARK方法消除线性时不定常常微分方程的降阶
Eliminating Order Reduction on Linear, Time-Dependent ODEs with GARK Methods
摘要：
当应用于刚性的线性微分方程与不定常强迫，龙格-库塔法可以表现出收敛速度低于经典的顺序条件理论的预测。通常，这种降阶现象是通过使用昂贵的、具有高阶阶的全隐式龙格-库塔法或满足附加阶条件的专门格式来解决的。本文发展了一种灵活的方法，用一种全隐式方法来处理强迫，如保持基本格式的经典阶数，来扩充任意龙格-库塔法。我们的方法和分析是基于一般结构的可加性Runge-Kutta框架。数值实验使用对角隐式，全隐式，甚至显式龙格-库塔法证实，新的方法消除了考虑中的问题的类降阶，和基本方法实现其理论的收敛阶。
英文摘要：
When applied to stiff, linear differential equations with time-dependent forcing, Runge-Kutta methods can exhibit convergence rates lower than predicted by the classical order condition theory. Commonly, this order reduction phenomenon is addressed by using an expensive, fully implicit Runge-Kutta method with high stage order or a specialized scheme satisfying additional order conditions. This work develops a flexible approach of augmenting an arbitrary Runge-Kutta method with a fully implicit method used to treat the forcing such as to maintain the classical order of the base scheme. Our methods and analyses are based on the general-structure additive Runge-Kutta framework. Numerical experiments using diagonally implicit, fully implicit, and even explicit Runge-Kutta methods confirm that the new approach eliminates order reduction for the class of problems under consideration, and the base methods achieve their theoretical orders of convergence.
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