Two New Fourth-Order Three-Stage Symplectic Integrators

Abstract

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 ϕ 4 model. When the nonintegrable lattice ϕ 4 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.

Application of the Störmer—Verlet-Like Symplectic Method to the Wave Equation

“…[18] As variants of the gradient algorithms, two Störmer-Verlet-like (SVL) symplectic methods of fourth-order accuracies were proposed recently. [19] These are efficient in working out the class of special potentials involving the harmonic oscillator, mathematic pendulum and lattice models. In particular, they are very useful in solving the variational equations of Hamiltonian systems since the variational potentials similar to the harmonic oscillator potential are quadratic forms of the variational position variables.…”