Efficient numerical integrators for highly oscillatory dynamic systems based on modified Magnus integrator method

Based on the Magnus integrator method established in linear dynamic systems, an efficiently improved modified Magnus integrator method was proposed for the second-order dynamic systems with time-dependent high frequencies. Firstly, the second-order dynamic system was reformulated as the first-order system and the frame of reference was transferred by introducing new variables so that highly oscillatory behavior inherits from the entries in the meantime. Then the modified Magnus integrator method based on local linearization was appropriately designed for solving the above new form and some improvement also were presented. Finally, numerical examples show that the proposed methods appear to be quite adequate for integration for highly oscillatory dynamic systems including Hamiltonian systems problem with long time and effectiveness.