A Note on the Generating Function Method (Advances in Applied Mathematics and Mechanics, 2021)
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.