Hamiltonian systems are known to conserve the Hamiltonian function, which describes the energy evolution over time. Obtaining a numerical spatio-temporal scheme that accurately preserves the discretized Hamiltonian function is often a challenge. In this paper, the use of high order mimetic spatial schemes is investigated for the numerical solution of Hamiltonian equations. The mimetic operators are based on developing high order discrete analogs of the vector calculus quantities divergence and gradient. The resulting high order operators preserve the properties of their continuum ones, and are therefore said to mimic properties of conservation laws and symmetries. Symplectic fourth order schemes are implemented in this paper for the time integration of Hamiltonian systems. A theoretical framework for the energy preserving nature of the resulting schemes is also presented, followed by numerical examples.

翻译：哈密顿系统以守恒哈密顿函数而著称，该函数描述了能量随时间的演化。获得能够精确保持离散化哈密顿函数的数值时空格式往往是一项挑战。本文研究了高阶模拟空间格式在哈密顿方程数值求解中的应用。模拟算子基于向量微积分量散度和梯度的高阶离散模拟。所得到的高阶算子保持其连续算子的性质，因此被认为模拟了守恒律和对称性。本文采用四阶辛格式进行哈密顿系统的时间积分，并提出了保持能量性质的数值格式的理论框架，随后给出了数值算例。