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

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