TY - JOUR
T1 - A Note on the Generating Function Method
AU - Bo , Yonghui
AU - Cai , Wenjun
AU - Wang , Yushun
JO - Advances in Applied Mathematics and Mechanics
VL - 4
SP - 982
EP - 1004
PY - 2021
DA - 2021/04
SN - 13
DO - http://doi.org/10.4208/aamm.OA-2020-0286
UR - https://global-sci.org/intro/article_detail/aamm/18760.html
KW - Hamiltonian systems, generating function methods, symplectic methods.
AB - <p style="text-align: justify;">The generating function method plays an important role in the construction of symplectic methods and closely depends on different generating functions. The three typical generating functions are widely applied in practical computations. This paper follows the general framework of the generating function method proposed by Feng Kang to produce a simple generating function with parameterized coordinates. This new generating function is more practical and covers the three typical ones by fixing the parameter to specific values. The relationship between symplectic transformation and new generating function and the Hamilton-Jacobi equation are discussed. A new family of arbitrary high-order symplectic methods with free parameter is obtained. Through the composition of the obtained low-order symplectic method, we derive another new class of any high-order symmetric symplectic methods with free parameter. These parametric symplectic methods will have more freedom of adjustment to design integrators which preserve energy or non-quadratic invariants. Computational examples illustrate the effectiveness of the proposed methods.</p>