On the Arbitrarily Long-Term Stability of Conservative Methods
On the Arbitrarily Long-Term Stability of Conservative Methods
复制标题
论保守方法的任意长期稳定性
DOI：
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发表时间：
2016
影响因子：
2.9
通讯作者：
Jean
中科院分区：
文献类型：
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作者：
A. Wan;Jean
我们展示了自主 ODE 保守方法的任意长期稳定性。给定具有守恒量的自治 ODE 系统，如果守恒量的原像具有有界局部有限邻域，则当均匀时间步长足够小时，任何具有均匀有界位移性质的保守方法的全局误差始终是有界的。在有限精度的机器上，全局误差仍然是有界的并且与时间无关，直到由机器精度和公差确定的任意大时间。主要结果是利用等连续离散守恒量的基本拓扑性质证明的。特别是，当离散守恒量不明确依赖于时间步长时，还可以使用平均恒等式来显示长期稳定性。给出的数值结果说明了长期稳定性结果。
We show the arbitrarily long-term stability of conservative methods for autonomous ODEs. Given a system of autonomous ODEs with conserved quantities, if the preimage of the conserved quantities possesses a bounded locally nite neighborhood, then the global error of any conservative method with the uniformly bounded displacement property is bounded for all time, when the uniform time step is taken suciently small. On nite precision machines, the global error still remains bounded and independent of time until some arbitrarily large time determined by machine precision and tolerance. The main result is proved using elementary topological properties for discretized conserved quantities which are equicontinuous. In particular, long-term stability is also shown using an averaging identity when the discretized conserved quantities do not explicitly depend on time steps. Numerical results are presented to illustrate the long-term stability result.