High asymptotic order numerical methods for highly oscillatory ODEs with large initial data

In this paper, we propose high asymptotic order numerical methods for solving highly oscillatory second order ODEs with large initial data, where the total energy of the system becomes unbounded as the oscillation frequency grows. The existing asymptotic-numerical solvers are especially designed for the classical energy bounded oscillatory equations, offering no insight into their performance with energy unbounded case. Based on the asymptotic expansion of the solution in the inverse powers of the oscillatory parameter, we propose an asymptotic numerical integrator to solve this class of highly oscillatory ODEs and discuss the computational efficiency for the case of polynomials. One numerical example is given to show the efficiency and accuracy of our proposed asymptotic-numerical solver.