Asymptotic-numerical solvers for highly oscillatory second-order differential equations

Zhongli Liu, Tianhai Tian, Hongjiong Tian

Research output: Contribution to journalArticleResearchpeer-review

4 Citations (Scopus)


In this paper, we propose an approach of combination of asymptotic and numerical techniques to solve highly oscillatory second-order initial value problems. 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 τ=ωt. The truncation with the first few terms of the expansion results in a very effective method of discretizing the highly oscillatory differential equation and becomes more accurate when the oscillatory parameter increases. Numerical examples show that our proposed asymptotic-numerical solver is efficient and accurate for highly oscillatory problems.

Original languageEnglish
Pages (from-to)184-202
Number of pages19
JournalApplied Numerical Mathematics
Publication statusPublished - 1 Mar 2019


  • Asymptotic expansion
  • Highly oscillatory problem
  • Modulated Fourier expansion
  • Multiple time scales
  • Second-order ordinary differential equation

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