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arXiv 1804.04553math.NAcs.NA
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关于在光滑非均匀网格上多步方法零稳定性的问题

On the zero-stability of multistep methods on smooth nonuniform grids

Lund University（隆德大学）；Eötvös Loránd University（罗兰大学）

Gustaf Söderlind, Imre Fekete, István Faragó

备注：
17 pages, Keywords: Initial value problems, linear multistep methods, BDF methods, zero stability, nonuniform grids, variable step size, convergence
发表信息：
BIT Numer Math 58 (2018) 1125-1143
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AI总结：

本文研究了在光滑非均匀网格上多步方法的零稳定性问题，提出了一种基于平滑变形映射的模型，并证明了在满足特定步长条件的情况下，强稳定的多步方法在足够大的N时具有零稳定性。

摘要：

为了保证收敛性，线性多步方法必须满足零稳定性。尽管常步长理论在1950年代已建立，但在非均匀网格上的零稳定性研究仍不够深入。本文探讨了在紧致区间和光滑非均匀网格上的零稳定性。在实际计算中，可以通过平滑的小步长变化来实现步长控制。所得到的网格{t_n}_{n=0}^N可以建模为等距网格在平滑变形映射下的像，即t_n = Φ(τ_n)，其中τ_n = n/N，且映射Φ单调递增，满足Φ(0)=0和Φ(1)=1。该模型在应用于光滑问题时，对于任何固定阶数的方法在其渐进行为下都成立，因为此时步长由光滑的主误差函数决定，该函数决定了Φ，而容差要求决定了N。给定任何强稳定的多步方法，存在一个N*，使得当N > N*时，该方法在Φ∈C^2[0,1]的情况下具有零稳定性。因此，所有满足相邻步长满足h_n/h_{n-1} = 1 + O(N^{-1})的非均匀网格上都具有零稳定性。结果通过BDF型方法进行了示例说明。

英文摘要：

In order to be convergent, linear multistep methods must be zero stable. While constant step size theory was established in the 1950's, zero stability on nonuniform grids is less well understood. Here we investigate zero stability on compact intervals and smooth nonuniform grids. In practical computations, step size control can be implemented using smooth (small) step size changes. The resulting grid $\{t_n\}_{n=0}^N$ can be modeled as the image of an equidistant grid under a smooth deformation map, i.e., $t_n = Φ(τ_n)$, where $τ_n = n/N$ and the map $Φ$ is monotonically increasing with $Φ(0)=0$ and $Φ(1)=1$. The model is justified for any fixed order method operating in its asymptotic regime when applied to smooth problems, since the step size is then determined by the (smooth) principal error function which determines $Φ$, and a tolerance requirement which determines $N$. Given any strongly stable multistep method, there is an $N^*$ such that the method is zero stable for $N>N^*$, provided that $Φ\in C^2[0,1]$. Thus zero stability holds on all nonuniform grids such that adjacent step sizes satisfy $h_n/h_{n-1} = 1 + \mathrm O(N^{-1})$ as $N\rightarrow\infty$. The results are exemplified for BDF-type methods.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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