Structure-Preserving Algorithms with Uniform Error Bound and Long-time Energy Conservation for Highly Oscillatory Hamiltonian Systems

Structure-preserving algorithms and algorithms with uniform error bound have constituted two interesting classes of numerical methods. In this paper, we blend these two kinds of methods for solving nonlinear systems with highly oscillatory solution, and the blended algorithms inherit and respect the advantage of each method. Two kinds of algorithms are presented to preserve the symplecticity and energy of the Hamiltonian systems, respectively. Long time energy conservation is analysed for symplectic algorithms and the proposed algorithms are shown to have uniform error bound in the position for the highly oscillatory structure. Moreover, some methods with uniform error bound in the position and in the velocity are derived and analysed. Two numerical experiments are carried out to support all the theoretical results established in this paper by showing the performance of the blended algorithms.