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

## TL;DR

This paper introduces a meshless numerical method combining HDMR and Fourier HC approximation to efficiently solve high-dimensional sine-Gordon equations with stability and accuracy.

## Contribution

It proposes a novel implicit meshless scheme using Fourier HDMR-HC to handle high-dimensional SGEs with sparse matrices and large time steps.

## Key findings

- Stable large time-step solutions achieved
- Reduced number of nodes needed for accuracy
- Effective simulation of high-dimensional SGEs

## Abstract

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.