In this paper, we consider highly oscillatory second-order differential equations $$\ddot{x}(t)+\Omega ^2x(t)=g(x(t))$$
 with a single frequency confined to the linear part, and $$\Omega $$
 is singular. It is known that the asymptotic-numerical solvers are an effective approach to numerically solve the highly oscillatory problems. Unfortunately, however, the existing asymptotic-numerical solvers fail to apply to the highly oscillatory second-order differential equations when $$\Omega $$
 is singular. We propose an efficient improvement on the existing asymptotic-numerical solvers, so that the asymptotic-numerical solvers can be able to solve this class of highly oscillatory ordinary differential equations. The error estimation of the asymptotic-numerical solver is analyzed and nearly conservation of the energy in the Hamiltonian case is proved. Two numerical examples including the Fermi–Pasta–Ulam problem are implemented to show the efficiency of our proposed methods.

Xiaoxue Sa

Papers

Asymptotic-numerical solvers for highly oscillatory ordinary differential equations and Hamiltonian systems