常微分方程数值解中舍入误差积累的定量研究
A quantitative investigation into the accumulation of rounding errors in the numerical solution of ODEs
AI总结：
通过理论分析和数值实验，研究了确定性ODE求解器中舍入误差的随机性积累及其对长期解发散行为的影响。
摘要：
我们研究了一些用于常微分方程（ODE）系统的确定性求解器的数值舍入误差。我们证明，舍入误差的积累导致解本质上是随机的，并且我们得到了轨迹作为时间、步长和计算机数值精度函数的理论分布。我们特别考虑了放大舍入误差影响的系统，使得在长时间内解表现出发散行为。通过使用不同的时间步长值进行多次重复，我们在数值上观察到了理论上预测的随机分布。我们主要关注显式欧拉法和RK4方法，但也简要考虑了更复杂的算法，如隐式求解器VODE和RADAU5。
英文摘要：
We examine numerical rounding errors of some deterministic solvers for systems of ordinary differential equations (ODEs). We show that the accumulation of rounding errors results in a solution that is inherently random and we obtain the theoretical distribution of the trajectory as a function of time, the step size and the numerical precision of the computer. We consider, in particular, systems which amplify the effect of the rounding errors so that over long time periods the solutions exhibit divergent behaviour. By performing multiple repetitions with different values of the time step size, we observe numerically the random distributions predicted theoretically. We mainly focus on the explicit Euler and RK4 methods but also briefly consider more complex algorithms such as the implicit solvers VODE and RADAU5.
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