A New Area-Preserving Geometric Numerical Algorithm of Nonlinear Systems

Abstract

Geometric numerical integration is a numerical algorithm that preserves the inherent geometric properties of the system. However, most traditional numerical algorithms do not take into account the characteristic, resulting in deviations in some properties during the long-term numerical discretization process. To overcome these issues, advanced numerical algorithms with invariant properties are required. Thus, this work reports the area-preserving performance of the Lie derivative geometric numerical algorithm in the Hamiltonian system. First, we propose a numerical discretization scheme of two-dimensional Hamiltonian systems via the Lie derivative method, and the convergence of the proposed algorithm is demonstrated. Then, the Jacobian matrix of numerical iterative algorithm is implemented, the absolute error, the mean squared error and the cumulative total error of area mapping are derived, respectively. Four criteria of the algorithm for area preservation are given. Furthermore, numerical calculation of two examples confirms that the solution errors over long times is quite satisfactory. Moreover, the cat-face mapping is achieved to prove the quantitative area-preserving.