Numerical meshless solution of high-dimensional sine-Gordon equations via Fourier HDMR-HC approximation
Numerical meshless solution of high-dimensional sine-Gordon equations via Fourier HDMR-HC approximation
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DOI：
10.1007/s10910-019-01030-3
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发表时间：
2019-05
影响因子：
1.7
通讯作者：
Xin Xu;Xiaopeng Luo;H. Rabitz
中科院分区：
文献类型：
--
作者：
Xin Xu;Xiaopeng Luo;H. Rabitz
本文提出了一种隐式时间步进无网格方案，通过结合高维模型表示（HDMR）和傅里叶双曲交叉（HC）近似来求解高维正弦-戈登方程（SGE）的数值解。为了保证隐式时间步进方案相关系数矩阵的稀疏性，首先将整个域划分为一组子域，每个子域中的高维相关导数可以分别通过傅里叶HDMR-HC近似来近似。所提出的方法允许稳定的大时间步长和相对较少的节点数量且具有令人满意的精度。数值例子表明，所提出的方法对于模拟高维 SGE 非常有吸引力。
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.