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Closed-form modified Hamiltonians for integrable numerical integration schemes

机译：可集成数值集成方案的封闭式修改哈密顿人

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Modified Hamiltonians are used in the field of geometric numerical integration to show that symplectic schemes for Hamiltonian systems are accurate over long times. For nonlinear systems the series defining the modified Hamiltonian usually diverges. In contrast, this paper constructs and analyzes explicit examples of nonlinear systems where the modified Hamiltonian has a closed-form expression and hence converges. These systems arise from the theory of discrete integrable systems. We present cases of one-and two-degrees symplectic mappings arising as reductions of nonlinear integrable lattice equations, for which the modified Hamiltonians can be computed in closed form. These modified Hamiltonians are also given as power series in the time step by Yoshida's method based on the Baker-Campbell-Hausdorff series. Another example displays an implicit dependence on the time step which could be of relevance to certain implicit schemes in numerical analysis. In light of these examples, the potential importance of integrable mappings to the field of geometric numerical integration is discussed.
机译：修改后的哈密顿人员用于几何数字集成领域，以表明Hamiltonian系统的杂项方案长时间是准确的。对于非线性系统，该系列定义了改进的汉密尔顿人通常发生差异。相反，本文构建和分析了改进的Hamiltonian具有闭合形式表达和因此收敛的非线性系统的明确示例。这些系统来自离散可积系统的理论。我们提出了一个和两个辛映射的情况，以减少非线性可排现的晶格方程式，修改后的哈密顿人可以以封闭形式计算。这些修改后的哈密顿人员也作为基于Baker-Campbell-Hausdorff系列的Yoshida的方法的时间步骤给出了电力系列。另一个示例显示了对数值分析中某些隐式方案的时间步骤的隐式依赖性。鉴于这些示例，讨论了可积映射对几何数值集成领域的潜在重要性。

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