作者:孙阳，宋琳琳，艾晓辉
Authors:SUN Yang,SONG Linlin,AI Xiaohui摘要:针对具有初边值问题的二维波动方程,提出了一种数值求解该方程的高阶紧致显式有限差分格式。 首先 ,根据相关文献对导数的离散近似,得到周期边界条件下的六阶紧致差分格式 。其次,在空间方向上,边界节点导数项利用原方程代入的方法进行计算,而内部节点的导数项利用六阶紧致差分公式近似,使空间精度达到六阶 。 同时 ,在时间方向上,利用泰勒级数展开公式、原方程代入以及中心差分公式推导出时间层的二阶精度差分格式,为了将整体上的时间精度由二阶提高至四阶,采用外推算法实现时间层的高阶近似 。再次,再利用傅里叶分析法对该格式的稳定性进行分析,得到在此精度下的稳定性条件,即 | a |λ ∈ [0 , 1/2(7/6)^1/2] 。最后 ,通过数值实验验证了所提出的 HOCE(6 ,4)格式的高效性和准确性。

Abstract:In this paper, a high-order compact explicit finite difference scheme is proposed to numerically solve two-dimensional
\nwave equations with initial boundary value problems. First of all, according to the discrete approximation of derivatives in the existing literature, a sixth-order compact difference scheme with periodic boundary conditions is obtained. Then, in the spatial direction, the derivative term of the boundary node is calculated by substituting the original equation, and the derivative term of the internal node is approximated by the sixth order compact difference formula, so that the spatial accuracy can reach the sixth order. For the time direction, the second-order accuracy difference scheme of the time layer is derived by using the Taylor series expansion formula, the original equation substitution and the central difference formula. In order to improve the overall time accuracy from the second order to the fourth order, the Richardson extrapolation method is used to realize the high-order approximation of the time layer. Furthermore the
\n1 7
\nstability of the scheme is analyzed by Fourier analysis method and the stability condition is | a | λ ∈ [0 , ]. Finally, the
\nefficiency and accuracy of the proposed HOCE(6 ,4) scheme are verified by numerical experiments.


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