The Study Of Several Force Gradient Symplectic Algorithms

A symplectic integrator, with the conservation of the symplectic structure of a Hamiltonian system and no secular change in the energy errors, has become a hot research topic.It should be emphasized that beyond second order any scheme does not avoid using some negative time coefficients. For the natural splitting of a Hamiltonian system into kinetic energy and potential energy, we construct two new optimal third order force-gradient symplectic algorithms, in each of which the norm of fourth-order truncation errors is minimized. They are both not explicitly superior to their no-optimal counterparts in the numerical stability and the topology structure-preserving, but they are in the accuracy of energy on classical problems and in one of the energy eigenvalues for one-dimensional time-independent Schrodinger equations. In particular, they are much better than the optimal third-order non-gradient symplectic method. They also have an advantage over the fourth-order non-gradient symplectic integrator. After then, this paper provides two new fourth-order force gradient symplectic integrators, each of which is obtained from a symmetric product of two identied optimal third-order force gradient symplectic integrators obtained above. They are both greatly superior to the fourth-order non-gradient symplectic integrator of Forest and Ruth in the accuracy of either energy on chaotic perturbed Kepler problems or the energy eigenvalues for one-dimensional Schrodinger equations. So are they to the known optimal fourth-order force gradient symplectic scheme.Two new fourth-order symplectic integrators are specifically designed for some Hamiltonian systems. Each of them contains three stages in every integration step, and looks like a usual non-gradient second-order symplectic integrator in the formal expression. When the harmonic oscillator, mathematical pendulum and lattice cpφ4 are taken as three physical models, numerical comparisons show that the new methods have a great advantage over the second-order 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 even are superior or almost equivalent to a fourth-order seven-stage force gradient symplectic integrator of Chin. They are particularly suitable for solving the variational equations of these systems. As an application, one of the new integrators and a good chaos indicator like the method of Lyapunov indicators or the fast Lyapunov indicators are together used to provide insight into the dynamics of the latticeφ4 systems, including a transition to chaos on the variation of single parameter and the structure of associated initial condition sets for chaos and regular motions.