Qualitative properties of modi"ed equations
Qualitative properties of modi"ed equations
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修正方程的定性性质
DOI：
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
A. Stuart
中科院分区：
文献类型：
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作者：
O. Gonzalez;D. Higham;A. Stuart
假设对光滑常微分方程组应用一个r阶相容的一步数值方法。对于任意大于1的整数m，该方法可以表示为rCm阶，作为对某个修正的艾德方程的近似.如果方法和系统具有特定的定性性质，则确定修改后的艾德方程是否继承该性质是重要的。本文介绍了一种证明改进的艾德方程继承了方法和基础系统的定性性质的技术。该技术使用了一个简单的矛盾参数适用于任意一步的方法，并不依赖于相关的幂级数展开的详细结构。因此，结论适用于，但不限于，龙格库塔方法的情况下。新方法统一和推广了这类结果，给出了积分保持、可逆性、定点继承、Hamilton问题和体积保持的结果.该技术也适用于当系统具有积分时，该方法不精确地保留积分，但阶数大于r。最后，考虑梯度系统和梯度数值方法具有的全局性质，得到了一个否定的结果，该全局性质不为相应的修正艾德方程所共有.
Suppose that a consistent one-step numerical method of order r is applied to a smooth system of ordinary differential equations. Given any integer m > 1, the method may be shown to be of order rC m as an approximation to a certain modi"ed equation. If the method and the system have a particular qualitative property then it is important to determine whether the modi"ed equations inherit this property. In this article, a technique is introduced for proving that the modi"ed equations inherit qualitative properties from the method and the underlying system. The technique uses a straightforward contradiction argument applicable to arbitrary one-step methods and does not rely on the detailed structure of associated power series expansions. Hence the conclusions apply, but are not restricted, to the case of Runge–Kutta methods. The new approach uni"es and extends results of this type that have been derived by other means: results are presented for integral preservation, reversibility, inheritance of "xed points, Hamiltonian problems and volume preservation. The technique also applies when the system has an integral that the method preserves not exactly, but to order greater than r. Finally, a negative result is obtained by considering a gradient system and gradient numerical method possessing a global property that is not shared by the associated modi"ed equations.