Symmetric Adaptive Higher-Order Energy-Preserving Methods for a Charged Particle System and Guiding Center System

Abstract

We propose higher-order adaptive energy-preserving methods for a charged particle system and a guiding center system. The higher-order energy-preserving methods are symmetric and are constructed by composing the second-order energy-preserving methods based on the averaged vector field. In order to overcome the energy drift problem that occurs in the energy-preserving methods based on the average vector field, we develop two adaptive algorithms for the higher-order energy-preserving methods. The two adaptive algorithms are developed based on using variable points of Gauss–Legendre’s quadrature rule and using two different stepsizes. The numerical results show that the two adaptive algorithms behave better in phase portrait and energy conservation than the Runge–Kutta methods. Moreover, it is shown that the energy errors obtained by the two adaptive algorithms can be bounded by the machine precision over long time and do not show energy drift.