二维线性双曲型方程Neumann边值问题的紧交替方向隐格式

doi:10.12286/jssx.2019.3.266

计算数学 ›› 2019, Vol. 41 ›› Issue (3): 266-294.DOI: 10.12286/jssx.2019.3.266

二维线性双曲型方程Neumann边值问题的紧交替方向隐格式

盛秀兰¹, 赵润苗², 吴宏伟²

1. 江苏开放大学, 南京 210036;
2. 东南大学数学学院, 南京 210096

收稿日期:2017-09-26 出版日期:2019-09-15 发布日期:2019-08-21
通讯作者: 曹学年,Email:cxn@xtu.edu.cn
基金资助:
国家自然科学基金项目（11671081）和江苏开放大学“十三五”规划课题（16SSW-Y-009）资助.

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盛秀兰, 赵润苗, 吴宏伟. 二维线性双曲型方程Neumann边值问题的紧交替方向隐格式[J]. 计算数学, 2019, 41(3): 266-294.

Sheng Xiulan, Zhao Runmiao, Wu Hongwei. A HIGH ORDER DIFFERENCE SCHEME FOR TWO-DIMENSIONAL LINEAR HYPERBOLIC EQUATION WITH NEUMANN BOUNDARY CONDITIONS[J]. Mathematica Numerica Sinica, 2019, 41(3): 266-294.

A HIGH ORDER DIFFERENCE SCHEME FOR TWO-DIMENSIONAL LINEAR HYPERBOLIC EQUATION WITH NEUMANN BOUNDARY CONDITIONS

Sheng Xiulan¹, Zhao Runmiao², Wu Hongwei²

1. Jiangsu Open University, Nanjing 210036, China;
2. School of Mathematics, Southeast University, Nanjing 210096, China

Received:2017-09-26 Online:2019-09-15 Published:2019-08-21

对二维Neumann边界条件的线性双曲型方程建立了紧交替方向的隐格式.利用方程和边界条件得到在空间上的三阶与五阶导数的边界值，进而在内点、边界内点和边界角点分别建立9点、6点和4点紧差分格式；通过引进新的范数和L₂范数估计L_∞范数；借助能量估计、Gronwall不等式和Schwarz不等式等技巧，详细分析了差分格式在无穷范数下关于时间和空间分别为二阶和四阶收敛性，并给出了稳定性结果；通过数值算例，验证了理论分析结果.

关键词： 线性双曲方程, 紧差分格式, 收敛性, 稳定性, 高精度

A high order difference scheme is established for two-dimensional linear hyperbolic equation with Neumann boundary conditions. The third and fifth derivatives of solution at the boundary can be got by using the boundary conditions and the equation, then the nine points, six points and four points compact difference schemes are respectively established at the inner points of the region, inner points and corner points of the boundary by using the finite difference method. To obtain the convergence and stability of the numerical solution in maximum norm, a new norm is introduced to estimate maximum norm. Then two priori estimates of the difference scheme are shown and convergence and stability are derived. The convergence order of the difference scheme in maximum norm is O(τ² + h⁴) where tau and h are temporal and spatial step size, respectively. Some numerical examples illustrate the convergence of the high order difference schemes presented in this paper.

Key words: Linear hyperbolic equation, Compact difference scheme, Convergence, Stability, High order accuracy

MR(2010)主题分类:

65M06
65M12

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摘要

二维线性双曲型方程Neumann边值问题的紧交替方向隐格式

A HIGH ORDER DIFFERENCE SCHEME FOR TWO-DIMENSIONAL LINEAR HYPERBOLIC EQUATION WITH NEUMANN BOUNDARY CONDITIONS

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