Birkhoff normal form for splitting methods applied to semilinear Hamiltonian PDEs. Part II. Abstract splitting

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• 作者：Erwan Faou ; Beno?t Gr¨¦bert and Eric Paturel
• 关键词：Mathematics Subject Classification (2000) 65P10 ; 37M15 ; 37J40
• 刊名：Numerische Mathematik
• 出版年：2010
• 出版时间：January, 2010
• 年：2010
• 卷：114
• 期：3
• 页码：459-490
• 全文大小：363.7 KB

文摘

We consider Hamiltonian PDEs that can be split into a linear unbounded operator and a regular non linear part. We consider abstract splitting methods associated with this decomposition where no discretization in space is made. We prove a normal form result for the corresponding discrete flow under generic non resonance conditions on the frequencies of the linear operator and on the step size, and under a condition of zero momentum on the nonlinearity. This result implies the conservation of the regularity of the numerical solution associated with the splitting method over arbitrary long time, provided the initial data is small enough. This result holds for rounded numerical schemes avoiding at each step possible high frequency energy drift. We apply these results to nonlinear Schr?dinger equations as well as the nonlinear wave equation.