When applied to stiff, linear differential equations with time-dependent forcing, Runge-Kutta methods can exhibit convergence rates lower than predicted by 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.
翻译:龙格-库塔方法在应用到具有基于时间的强制力的硬性线性差异方程式时,其趋同率可能低于古典秩序条件理论的预测。通常,这种减少订单现象是通过使用昂贵的、完全隐含的、具有高档顺序的龙格-库塔方法或满足额外秩序条件的专门办法来解决的。这项工作发展了一种灵活的办法,即增加任意的龙格-库塔方法,采用完全隐含的方法处理压力,例如维持基础办法的古典秩序。我们的方法和分析以一般结构添加剂龙格-库塔框架为基础。使用直线、完全隐含甚至直线的龙格-库塔方法进行的数值实验证实,新办法消除了审议中各类问题的减少订单,基本方法实现了理论趋同。