Testing a New Conservative Method for Solving the Cauchy Problem for Hamiltonian Systems on Test Problems

Abstract

A new numerical method for solving the Cauchy problem for Hamiltonian systems is tested in detail as applied to two benchmark problems: the one-dimensional motion of a point particle in a cubic potential field and the Kepler problem. The global properties of the resulting approximate solutions, such as symplecticity, time reversibility, total energy conservation, and the accuracy of numerical solutions to the Kepler problem are investigated. The proposed numerical method is compared with three-stage symmetric symplectic Runge–Kutta methods, the discrete gradient method, and nested implicit Runge–Kutta methods.