  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.
