General linear methods
General linear methods
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DOI：
10.1017/s0962492906220014
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发表时间：
2006-01-01
期刊：
影响因子：
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通讯作者：
Butcher, J. C.
中科院分区：
文献类型：
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作者：
Butcher, J. C.
一般线性方法作为多步多值方法，是线性多步方法和龙格-库塔方法的自然推广。这项调查包含了对传统方法的讨论和对一般线性类型概括的动机。文中介绍了新方法的表述及其一致性、稳定性和收敛的基本性质。一般线性方法的阶必须从新的角度来看待，并展示了如何使用代数结构(等价于B级数)来表示给定阶的条件。新方法的线性和非线性稳定性使经典方法的理论得到了全面的阐述，并概述了已有的结果。最近已经介绍了一些亚家族，并对其中一些进行了详细的考虑。这尤其适用于具有“固有Runge-Kutta稳定性”性质的方法。这些方法似乎有可能产生有用和有效的方法，并概述了它们在实际实施方面的一些进展。最后，在一个足够广泛的范围内讨论了稳定性函数与方法阶数之间的关系，包括一般的线性方法和多重导数方法，如Obreshkov方法。由于Ehle，Daniel-Moore和Dahlquist(第二个障碍)的经典障碍都符合一个共同的模式，这些都是在一般背景下探讨的。
General linear methods, as multistage multivalue methods, are the natural generalizations of linear multistep and Runge-Kutta methods. This survey contains a discussion of the traditional methods and a motivation for the general linear type of generalization. The new methods are introduced in terms of their formulation and the basic properties of consistency, stability and convergence. The order of general linear methods has to be looked at from a new point of view and it is shown how to use an algebraic structure (equivalent to B-series) to express conditions for a given order. Linear and non-linear stability for the new methods brings the theories for the classical methods into a comprehensive formulation and known results are outlined. Recently a number of subfamilies have been introduced and some of these are considered in detail. This applies in particular to methods with the property known as ‘inherent Runge-Kutta stability’. These seem to have prospects of yielding useful and efficient methods, and some progress towards their practical implementation is outlined. Finally, the relationship between stability functions and order of methods is discussed in a setting wide enough to include general linear methods as well as multiderivative methods, such as Obreshkov methods. The classical barriers due to Ehle, Daniel-Moore and Dahlquist (second barrier) all fit into a common pattern and these are explored in a general setting.