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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2015, Vol. 32 ›› Issue (4): 577-589.doi: 10.3969/j.issn.1005-3085.2015.04.011

• • 上一篇    下一篇

一类非局部粘性水波模型的数值格式

张  俊1,   李物兰2   

  1. 1- 贵州财经大学数学与统计学院，贵阳  550025
    2- 温州医科大学信息科学与计算机工程学院，温州 325035
  • 收稿日期:2014-05-04 接受日期:2014-12-02 出版日期:2015-08-15 发布日期:2015-10-15
  • 基金资助:
    国家自然科学基金 (11461012)；温州医科大学科研课题项目 (QTJ11014)；浙江省教育厅科研资助项目 (Y201328047).

Numerical Approximation to a Shallow Wave Model with a Nonlocal Viscous

ZHANG Jun1,   LI Wu-lan2   

  1. 1- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550025
    2- College of Information Science and Computer Engineering, Wenzhou Medical University, Wenzhou 325035
  • Received:2014-05-04 Accepted:2014-12-02 Online:2015-08-15 Published:2015-10-15
  • Supported by:
    The National Natural Science Foundation of China (11461012); the Scientific Research Development Fund of Wenzhou Medical University (QTJ11014); the Research Project of Department Edu-cation of Zhejiang Province (Y201328047).

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摘要： 本文主要讨论带非局部粘性项水波方程的数值方法．我们建立了一种求解这类粘性水波方程的数值方案．该方案有效解决了非局部粘性项与非线性项的离散问题．所提的格式包括对$\alpha$阶分数阶项的$2-\alpha$阶格式和对非线性项的线性化处理的混合格式．我们证明了这种格式是无条件稳定的，并得出线性Crank-Nicolson加$2-\alpha$ 格式的收敛阶是${O}(\Delta t^{\frac{3}{2}}+N^{1-m})$的结论．一系列的数值例子验证了理论证明的正确性．最后，我们用所提数值格式研究了粘性水波方程的渐近衰减率，并讨论了各种参数项对衰减率的影响．

关键词: 分数阶, 无条件稳定, 有限差分法, 谱方法, 衰减率

Abstract:

We focus on the numerical investigation of a water wave model with a nonlocal viscous dispersive term. We construct and analyze a schema to numerically solving the nonlocal water wave model. The key for the success consists in a particular combination of the treatments for the nonlocal dispersive term and nonlinear convection term. The proposed methods employ a known $(2-\alpha)$-order schema for the $\alpha$-order fractional derivative and a mixed linearization of the nonlinear term. A rigorous analysis shows that the proposed schema is unconditionally stable, and the linearized  Crank-Nicolson plus $(2-\alpha)$--order schemes is ${O}( \Delta t^{\frac{3}{2}} +N^{1-m})$. A series of numerical examples is presented to confirm the theoretical prediction. Finally the proposed methods are used to investigate the asymptotical decay rate of the solutions of the nonlocal viscous wave equation, as well as the impact of different terms on this decay rate.

Key words: fractional order, unconditionally stable, finite difference methods, spectral methods, decay rate

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