Explicit K-symplectic-like algorithms for guiding center system

Abstract

In this paper, for the guiding center system, we propose a type of explicit K-symplectic-like methods by extending the original guiding center phase space and constructing new augmented Hamiltonians. The original guiding center phase space is extended by making several copies in order to make the guiding center Hamiltonian separable to variables. In the extended phase space, the augmented guiding center Hamiltonian can be numerically solved by a K-symplectic method through the splitting technique and the composition of some simpler subsystems. Meanwhile, a midpoint permutation constraint is imposed on the extended phase space. Numerical experiments are carried out for guiding center motions in different magnetic fields using different numerical methods, including K-symplectic-like algorithms, canonical symplectic algorithms, and higher order implicit Runge-Kutta methods. Results show that energy errors of K-symplectic-like methods are bounded within small intervals over a long time, defeating higher order implicit Runge-Kutta methods. For comparison, explicit K-symplectic-like methods exhibit higher computational efficiency than existing canonicalized symplectic methods of the same order. We also verify that permutation constraints are important for the numerical properties of explicit K-symplectic methods. Among them, the method with the midpoint permutation constraint behaves better in long-term energy conservation and the elimination of secular drift errors than the same method without any permutation. The permutation that imposes a constraint on the Hamiltonian behaves best in energy preservation.