Closed-form modified Hamiltonians for integrable numerical integration schemes
Closed-form modified Hamiltonians for integrable numerical integration schemes
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DOI：
10.1088/1361-6544/aad9ac
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发表时间：
2017-07
期刊：
影响因子：
1.7
通讯作者：
Shami A. M. Alsallami;Jitse Niesen;F. Nijhoff
中科院分区：
文献类型：
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作者：
Shami A. M. Alsallami;Jitse Niesen;F. Nijhoff
修正哈密顿量用于几何数值积分领域，证明了哈密顿系统的辛格式在长时间内是精确的。对于非线性系统，定义修正哈密顿量的级数通常是发散的。与此相反，本文构造并分析了非线性系统的显式例子，其中修正哈密顿量具有封闭形式并因此收敛。这些系统起源于离散可积系统理论。我们给出了由非线性可积格方程的约化而产生的1度和2度辛映射，其修正哈密顿量可以用封闭形式计算。在Baker-Campbell-Hausdorff级数的基础上，用Yoshida的方法给出了这些修正的哈密顿量在时间步长的幂级数。另一个例子显示了对时间步长的隐式依赖，这可能与数值分析中的某些隐式方案相关。根据这些例子，讨论了可积映射在几何数值积分领域的潜在重要性。
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.