Optimized third-order force-gradient symplectic algorithms

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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.

Two New Fourth-Order Three-Stage Symplectic Integrators

Li¹,

Wu²

2011

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A New Class of Scaling Correction Methods

Mei¹,

Wu²,

Liu³

2012

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When conventional integrators like Runge—Kutta-type algorithms are used, numerical errors can make an orbit deviate from a hypersurface determined by many constraints, which leads to unreliable numerical solutions. Scaling correction methods are a powerful tool to avoid this. We focus on their applications, and also develop a family of new velocity multiple scaling correction methods where scale factors only act on the related components of the integrated momenta. They can preserve exactly some first integrals of motion in discrete or continuous dynamical systems, so that rapid growth of roundoff or truncation errors is suppressed significantly.

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

Qiu¹,

Wu²

2013

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A fourth-order three-stage symplectic integrator similar to the second-order Störmer-Verlet method has been proposed and used before [Chin. Phys. Lett. 28 (2011) 070201; Eur. Phys. J. Plus 126 (2011) 73]. Continuing the work initiated in the publications, we investigate the numerical performance of the integrator applied to a one-dimensional wave equation, which is expressed as a discrete Hamiltonian system with a fourth-order central difference approximation to a second-order partial derivative with respect to the space variable. It is shown that the Störmer-Verlet-like scheme has a larger numerical stable zone than either the Störmer-Verlet method or the fourth-order Forest-Ruth symplectic algorithm, and its numerical errors in the discrete Hamiltonian and numerical solution are also smaller.