Numerical integration schemes are mandatory to understand complex behaviors of dynamical systems described by ordinary differential equations. Implementation of these numerical methods involve floating-point computations and propagation of round-off errors. This paper presents a new fine-grained analysis of round-off errors in explicit Runge-Kutta integration methods, taking into account exceptional behaviors, such as underflow and overflow. Linear stability properties play a central role in the proposed approach. For a large class of Runge-Kutta methods applied on linear problems, a tight bound of the round-off errors is provided. A simple test is defined and ensures the absence of underflow and a tighter round-off error bound. The absence of overflow is guaranteed as linear stability properties imply that (computed) solutions are non-increasing.

中文翻译：

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显式 Runge-Kutta 方法的舍入误差和异常行为分析

数值积分方案对于理解由常微分方程描述的动态系统的复杂行为是强制性的。这些数值方法的实现涉及浮点计算和舍入误差的传播。本文提出了一种新的对显式 Runge-Kutta 积分方法中舍入误差的细粒度分析，同时考虑了异常行为，例如下溢和上溢。线性稳定性特性在所提出的方法中起着核心作用。对于应用于线性问题的一大类 Runge-Kutta 方法，提供了舍入误差的严格界限。定义了一个简单的测试并确保没有下溢和更严格的舍入误差界限。保证没有溢出，因为线性稳定性属性意味着（计算的）解是非增加的。