Stepsize Conditions for Boundedness in Numerical Initial Value Problems
Stepsize Conditions for Boundedness in Numerical Initial Value Problems
复制标题
数值初值问题中的有界性步长条件
DOI：
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发表时间：
2009
影响因子：
2.9
通讯作者：
M. N. Spijker
中科院分区：
文献类型：
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作者：
W. Hundsdorfer;A. Mozartova;M. N. Spijker
对于Runge-Kutta方法（RKM）、线性多步法（LLM）和一般线性方法（GLM），在文献中，人们非常关注由总变差递减、强稳定性保持和单调性等术语表示的特殊非线性稳定性要求。保证这些性质的步长条件由Shu & Osher [J. Comput.物理、第77（1988）号决议，第77页。#24471;，并在随后的许多论文中。这些特殊的稳定性要求意味着数值方法的本质有界性，其中包括全变差有界性。不幸的是，对于许多众所周知的方法，上述特殊要求是违反的，所以不能以这种方式得出结论，该方法是（全变差）有界的。在本文中，我们专注于步长条件的有界性直接，而不是通过绕道上述特殊的稳定性。我们提出了一个通用的框架，以获得最佳的步长条件，保证实际RKM，LVMH，和GLM的有界性，从而推广上述特殊的稳定性的结果。
For Runge-Kutta methods (RKMs), linear multistep methods (LMMs), and classes of general linear methods (GLMs), much attention has been paid, in the literature, to special nonlinear stability requirements indicated by the terms total-variation-diminishing, strong stability preserving, and monotonicity. Stepsize conditions, guaranteeing these properties, were derived by Shu & Osher [J. Comput. Phys., 77 (1988), pp. 439-471] and in numerous subsequent papers. These special stability requirements imply essential boundedness properties for the numerical methods, among which the property of being total-variation-bounded. Unfortunately, for many well-known methods, the above special requirements are violated, so that one cannot conclude in this way that the methods are (total-variation-)bounded. In this paper, we focus on stepsize conditions for boundedness directly, rather than via the detour of the above special stability properties. We present a generic framework for deriving best possible stepsize conditions which guarantee boundedness of actual RKMs, LMMs, and GLMs, thereby generalizing results on the special stability properties mentioned above.