Integration of stiff mechanical systems by Runge-Kutta methods
Integration of stiff mechanical systems by Runge-Kutta methods
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DOI：
10.3929/ethz-a-004283507
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发表时间：
1993-11
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影响因子：
--
通讯作者：
C. Lubich
中科院分区：
文献类型：
--
作者：
C. Lubich
研究了刚性力学系统的数值积分，其中强势迫使运动保持接近流形。运动方程被写成一个具有小刚度参数ε的奇异摄动问题。光滑的解决方案，这样的系统的特点，在区别于高振荡的一般解决方案。隐式Runge-Kutta方法使用的步长大于100的近似光滑的解决方案，并得出精确的误差估计。当ω → 0时，刚性系统的龙格-库塔解收敛到相关约束系统的龙格-库塔解，该约束系统被表示为指数为3的微分代数方程。刚性初值问题的标准软件不能令人满意地工作在这里考虑的刚性系统。解释了失败的原因，并提出了补救措施。
The numerical integration of stiff mechanical systems is studied in which a strong potential forces the motion to remain close to a manifold. The equations of motion are written as a singular singular perturbation problem with a small stiffness parameter ɛ. Smooth solutions of such systems are characterized, in distinction to highly oscillatory general solutions. Implicit Runge-Kutta methods using step sizes larger than ɛ are shown to approximate smooth solutions, and precise error estimates are derived. As ɛ → 0, Runge-Kutta solutions of the stiff system converge to Runge-Kutta solutions of the associated constrained system formulated as a differential-algebraic equation of index 3. Standard software for stiff initial-value problems does not work satisfactorily on the stiff systems considered here. The reasons for this failure are explained, and remedies are proposed.