Two New Fourth-Order Three-Stage Symplectic Integrators

Two new fourth-order three-stage symplectic integrators are specifically designed for a family of Hamiltonian systems, such as the harmonic oscillator, mathematical pendulum and lattice φ⁴ model. When the nonintegrable lattice φ⁴ system is taken as a test model, numerical comparisons show that the new methods have a great advantage over the second-order Verlet symplectic integrators in the accuracy of energy, become explicitly better than the usual non-gradient fourth-order seven-stage symplectic integrator of Forest and Ruth, and are almost equivalent to a fourth-order seven-stage force gradient symplectic integrator of Chin. As the most important advantage, the new integrators are convenient for solving the variational equations of many Hamiltonian systems so as to save a great deal of the computational cost when scanning a lot of orbits for chaos.