Numerical meshless solution of high-dimensional sine-Gordon equations via Fourier HDMR-HC approximation

Xin Xu, Xiaopeng Luo, Herschel Rabitz

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


In this paper, an implicit time stepping meshless scheme is proposed to find the numerical solution of high-dimensional sine-Gordon equations (SGEs) by combining the high dimensional model representation (HDMR) and the Fourier hyperbolic cross (HC) approximation. To ensure the sparseness of the relevant coefficient matrices of the implicit time stepping scheme, the whole domain is first divided into a set of subdomains, and the relevant derivatives in high-dimension can be separately approximated by the Fourier HDMR-HC approximation in each subdomain. The proposed method allows for stable large time-steps and a relatively small number of nodes with satisfactory accuracy. The numerical examples show that the proposed method is very attractive for simulating the high-dimensional SGEs.

Original languageEnglish (US)
Pages (from-to)1683-1699
Number of pages17
JournalJournal of Mathematical Chemistry
Issue number7
StatePublished - Aug 15 2019

All Science Journal Classification (ASJC) codes

  • General Chemistry
  • Applied Mathematics


  • High dimensional model representation
  • Meshless methods
  • Sine-Gordon equations


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