A general symplectic scheme with three free parameters and its applications

Abstract

Abstract Symplectic schemes applied to Hamiltonian systems have prominent advantages for the preservation of qualitative properties of the flow. Three types of symplectic methods, which contain the symplectic Euler, implicit midpoint and Stormer–Verlet methods, are simplest and widely used in actual calculations. In this paper, we introduce a simple symplectic scheme with three free parameters, which covers these three methods and has the same behaviors as them. The symplecticity of this scheme is verified from partitioned Runge–Kutta methods and variational integrators. In addition, we get a second-order symplectic scheme with two free parameters and a symmetric symplectic scheme with a free parameter. By adjusting the parameter at each time step, we get a second-order energy and quadratic invariants preserving method. The effectiveness of all schemes is demonstrated by numerical tests.