On Symplectic and Multisymplectic Schemes for the KdV Equation
On Symplectic and Multisymplectic Schemes for the KdV Equation
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DOI：
10.1007/s10915-004-4634-6
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发表时间：
2005-10
影响因子：
2.5
通讯作者：
U. Ascher;R. McLachlan
中科院分区：
文献类型：
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作者：
U. Ascher;R. McLachlan
我们研究了著名的Korteweg-de Vries方程的一些辛方法和多辛方法，问题是这些方法提供的附加结构保持是否是为非线性守恒偏微分方程长时间积分提供高质量格式的关键。以二阶离散为中心，构造并研究了几种有趣的格式。我们的基本结论是，对于非线性守恒的KdV方程，设计非常稳定的守恒差分格式是可能的。在这类格式中，最好的方法是辛或多辛方法。半显式辛格式在许多情况下都是非常有效的。紧凑盒格式在确保近似解不会出现人为摆动方面是有效的。构造了一族盒格式，其中多辛盒格式是其中的一个重要成员，它们在粗时空网格上特别稳定
We examine some symplectic and multisymplectic methods for the notorious Korteweg–de Vries equation, with the question whether the added structure preservation that these methods offer is key in providing high quality schemes for the long time integration of nonlinear, conservative partial differential equations. Concentrating on second order discretizations, several interesting schemes are constructed and studied. Our essential conclusions are that it is possible to design very stable, conservative difference schemes for the nonlinear, conservative KdV equation. Among the best of such schemes are methods which are symplectic or multisymplectic. Semi-explicit, symplectic schemes can be very effective in many situations. Compact box schemes are effective in ensuring that no artificial wiggles appear in the approximate solution. A family of box schemes is constructed, of which the multisymplectic box scheme is a prominent member, which are particularly stable on coarse space–time grids