在Kahan的方法中使用香味来搜索保留的度量和积分

IF 4.6 Q2 MATERIALS SCIENCE, BIOMATERIALS
Geir Bogfjellmo, Elena Celledoni, Robert McLachlan, Brynjulf Owren, G. Quispel
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引用次数: 0

摘要

应用于二次微分方程的Kahan数值方法通常产生低维的可积映射,并且可以在更一般的情况下表现出保留测度和积分。基于离散达布多项式的计算机化方法最近被用于寻找这些度量和积分。然而,如果微分系统包含许多参数,这种方法可能导致高度复杂的结果,难以解释和分析。但在某些情况下,这种复杂性可以通过使用芳香系列大大降低。这些是由Chartier和Murua以及Iserles, Quispel和Tse独立引入的数学工具。我们为此目的开发了一种算法,并推导了Kahan映射具有可用芳族函数表示的保留测度和积分的必要条件。该方法成功的一个重要原因是矢量场到其芳族函数的映射的等方差。我们在许多例子上演示了该算法,与固定基(如单项式)相比,该算法的复杂性大大降低。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Using aromas to search for preserved measures and integrals in Kahan’s method
The numerical method of Kahan applied to quadratic differential equations is known to often generate integrable maps in low dimensions and can in more general situations exhibit preserved measures and integrals. Computerized methods based on discrete Darboux polynomials have recently been used for finding these measures and integrals. However, if the differential system contains many parameters, this approach can lead to highly complex results that can be difficult to interpret and analyse. But this complexity can in some cases be substantially reduced by using aromatic series. These are a mathematical tool introduced independently by Chartier and Murua and by Iserles, Quispel and Tse. We develop an algorithm for this purpose and derive some necessary conditions for the Kahan map to have preserved measures and integrals expressible in terms of aromatic functions. An important reason for the success of this method lies in the equivariance of the map from vector fields to their aromatic functions. We demonstrate the algorithm on a number of examples showing a great reduction in complexity compared to what had been obtained by a fixed basis such as monomials.
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来源期刊
ACS Applied Bio Materials
ACS Applied Bio Materials Chemistry-Chemistry (all)
CiteScore
9.40
自引率
2.10%
发文量
464
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