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

  • Xin XuEmail author
  • Xiaopeng Luo
  • Herschel Rabitz
Original Paper


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.


Sine-Gordon equations Meshless methods High dimensional model representation 



The authors X.X. and X.L. acknowledge support from the National Science Foundation (Grant No. CHE-1763198), and H.R. acknowledges support from the Templeton Foundation (Grant No. 52265).


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Authors and Affiliations

  1. 1.Department of ChemistryPrinceton UniversityPrincetonUSA
  2. 2.School of Management and EngineeringNanjing UniversityNanjingChina
  3. 3.Program in Applied and Computational MathematicsPrinceton UniversityPrincetonUSA

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