Author
Zhongli Liu
Bio: Zhongli Liu is an academic researcher from Shanghai Normal University. The author has contributed to research in topics: Asymptotic expansion & Initial value problem. The author has an hindex of 1, co-authored 1 publications receiving 1 citations.
Papers
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TL;DR: An asymptotic expansion of the solution is derived in inverse of powers of the oscillatory parameter, which develops on two time scales, a slow time t and a fast time ω t .
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TL;DR: In this paper , an asymptotic expansion for the oscillatory term of the solution of diffusion equation with time-like highly oscillatory forcing terms was derived, where each term can be computed at a lower cost.
TL;DR: In this paper, the authors proposed an efficient improvement on the existing asymptotic-numerical solvers, which can solve the class of highly oscillatory ordinary differential equations.
Abstract: 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.