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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[1] Dehghan M, Shokri A. A meshless method for numerical solution of a linear hyperbolic equation with variable coffcients in two space dimensional[J]. Numer. Methods Partial Differential Equations, 2009, 25:494-506.

[2] He Dongdong. An unconditionally stable spatial sixth-order CCD-ADI method for the two dimensionallinear telegraph equation[J]. Numer. Algorithms, 2016, 72(4):1103-1117.

[3] Mohanty R K, Jain M K. An unconditionally stable alternating direction implicit scheme for the two space dimensional linear hyperbolic equation[J]. Numer. Methods Partial Differential Equations, 2001, 17(6):684-688.

[4] Liu J, Tang K. A new unconditionally stable ADI compact scheme for the two-space dimensional linear hyperbolic equation[J]. Internat. J. Comput. Math., 2010, 87(10):2259-2267.

[5] Evans D J, Bulut H. The numerical solution of Burgers' equation by the alternating group explicit (age) method[J]. Internat. J. Comput. Math., 2003, 29(1):1289-1297.

[6] Hu Y Y, Liu H W. An unconditionally stable spline difference scheme for solving the second 2D linear hyperbolic equation[C]. Computer Modelling and Simulation International Conference, on (2010), Sanya China, 375-378.

[7] Mehdi Dehghan, Akbar Mohebbi. The combination of collocation, fnite difference, and multigrid methods for solution of the two-dimensional wave equation[J]. Numer. Methods Partial Differential Equations, 2008, 24(3):897-910.

[8] Mohanty R K. An operator splitting method for an unconditionally stable difference scheme for a linear hyperbolic equation with variable coefficients in two space dimensions[J]. Appl. Math. Comput., 2004, 152(3):799-806.

[9] Dehghan M, Ghesmati A. Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method[J]. Eng. Anal. Bound. Elem., 2010, 34(1):51-59.

[10] Pekmen B, Tezer-Sezgin M. Differential quadrature solution of hyperbolic telegraph equation[J]. J. Appl. Math., 2012, 2012, Article ID 924765, 18 pages.

[11] Ram Jiwari, Sapna Pandit, Mittal R C. A differential quadrature algorithm to solve the two dimensional linear hyperbolic telegraph equation with Dirichlet and Neumann boundary conditions[J]. Appl. Math. Comput., 2012, 218(13):7279-7294.

[12] Dehghan M, Ghesmati A. Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation[J]. Eng. Anal. Bound. Elem., 2010, 34(4):324-336.

[13] Heinz-Otto Kreiss, Anders Petersson N, Jacob Ystrom. Difference approximations of the Neumann problem for the second order wave equation[J]. SIAM J. Numer. Anal., 2004, 42(3):1292-1323.

[14] Appelo D, Petersson N A. A fourth-order accurate embedded boundary method for the wave equation[J]. SIAM J. Sci. Comput., 2012, 34(6):A2982-A3008.

[15] Gao Guang-Hua, Sun Zhi-Zhong. Compact difference schemes for heat equation with Neumann boundary conditions (Ⅱ)[J]. Numer. Methods Partial Differential Equations, 2013, (29):1459-1486.

[16] Sun Z Z. Compact difference schemes for heat equation with Neumann boundary conditions[J]. Numer. Methods Partial Differential Equations, 2010, 25(6):1320-1341.

[17] Zhou Y L. Application of discrete functional analysis to the fnite difference methods[M]. Interna-tional Academic Publishers, 1990, 8(1):49-65.

[18] 万正苏.带导数边界条件的线性双曲方程的一个二阶收敛格式[J].高等学校计算数学学报, 2002, 24(2):212-224.

[19] Li J, Sun Z Z, Zhao X. A three level linearized compact difference scheme for the Cahn-Hilliard equation[J]. Sci. China Math., 2012, 55(4):805-826.

[20] Liao H L, Sun Z Z. Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations[J]. Numer. Methods Partial Differential Equations, 2010, 26(1):37-60.

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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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